{VERSION 4 0 "DEC ALPHA UNIX" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 1 }{CSTYLE "" -1 25 " Courier" 0 1 0 0 0 1 2 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 20 0 0 0 0 0 1 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "helvetica" 1 16 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 276 "helvetica" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "fixed" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "fixed" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Map le Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "helvetica" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 264 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 256 67 "Singular and General Solutions of Al gebraic Differential Equations " }}{PARA 257 "" 0 "" {TEXT 258 52 "Exa mples and their analysis with the diffalg package" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT 257 16 "Evelyne Hubert \n" }} {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "P(x,y,y[1], `...`,y[n]) " "6#-%\"PG6'%\"xG%\"yG&F'6#\" \"\"%$...G&F'6#%\"nG" }{TEXT -1 76 " be a multivariate polynomial over a field of characteristic zero. We write " }{XPPEDIT 18 0 "`y'`=diff( y(x),x)" "6#/%#y'G-%%diffG6$-%\"yG6#%\"xGF+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 53 "A solution of the ordinary differential equatio n " }{XPPEDIT 18 0 "P(x,y,`y'`,`...`,y^`(n)`) = 0;" "6#/-%\"PG6'% \"xG%\"yG%#y'G%$...G)F(%$(n)G\"\"!" }{TEXT -1 18 " is said to be " }{TEXT 278 8 "singular" }{TEXT -1 20 " if it satisfies " }{XPPEDIT 18 0 "diff(P,y[n]) = 0;" "6#/-%%diffG6$%\"PG&%\"yG6#%\"nG\"\"!" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 356 "This is the definition that prevails since the work of Darboux (1870) on the subject. It is \+ nonetheless not equivalent to the original idea of Laplace (1772) and Lagarange (1774), for whom the singular solution of a first order dif ferential equation is a solution that cannot be obtained by substituti ng a numerical value to the arbitrary constant of the " }{TEXT 277 16 "general solution" }{TEXT -1 63 ". This non-equivallence will be illu strated in a next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 305 "Clairaut (1734) seems to have been the first to \+ come across the existence of singular solutions. In the equations, to \+ which he has given his name, the singular solution is an enveloppe of \+ the nonsingular solutions. We will illustrate that this need not to be the case. See also the introductory document :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT 280 81 "\"How to define the ge neral and the singular solution of a differential equation?\"" }{TEXT -1 7 " \non " }{TEXT 282 43 "http://www-sop.inria.fr/cafe/Evelyne.Hu bert" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 421 " We will give example of partial differential equations of first order \+ to show that their singular solutions exhibit similar properties as t he ones of first order differential equations, namely some enveloping \+ and limiting property. These are the only two possibilities for first \+ order. With the toboganning example, we shall show that these two case s can occur in a simple physical sytem according to the initial energy ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "We \+ will see, again with examples, the diversity of behaviour that appear \+ in higher order differential equations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 226 "In the section preparation polynomi al and minimal decomposition, we introduce the tools to make a first c lassification of the singular solutions according to their properties. For a complete explanation of these tools, look at" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 11 "E. Hubert, " }{TEXT 279 56 "Essential component of a differential algebraic equation" } {TEXT -1 77 ", Journal of Symbolic Computations (1999) volume 29 numbe r 4-5 pages 657-680." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 282 "These tools are used on three examples. The Universal \+ equation of Rubel, a Cahzy equation and the equation of the travelling wave of the KdV. In this latter we see how our tools can be used to d etermine the nature of the wave. The tools are implemented in Maple an d embedded in the " }{TEXT 281 7 "diffalg" }{TEXT -1 68 " package, a p ackage of differential algebra. See and download from:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT 283 45 "http://daisy.uwat erloo.ca:80/~ehubert/Diffalg" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(diffalg);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#7=%3Rosenfeld_GroebnerG%+belongs_toG%-delta_lead erG%1delta_polynomialG%'denoteG%,derivativesG%2differential_ringG%3dif ferential_spremG%.differentiateG%*equationsG%5essential_componentsG%0f ield_extensionG%(greaterG%,inequationsG%(initialG%3initial_conditionsG %.is_orthonomicG%'leaderG%6power_series_solutionG%7preparation_polynom ialG%.print_rankingG%%rankG%(reducedG%-reduced_formG%.rewrite_rulesG%) separantG%(versionG" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 19 "Clairaut' s equation" }}{PARA 0 "" 0 "" {TEXT -1 174 "Clairaut's work was concer ned with geometry. He first brought out this paradox that , for certai n differential equations, there exist solutions which are not embedded in the " }{TEXT 261 16 "general solution" }{TEXT -1 72 " - i.e. that \+ cannot be obtained by specialising the arbitrary constants." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "A class of diff erential equations is named after Clairaut. Clairaut gave the mean to \+ determine the " }{TEXT 262 19 "singular solutions " }{TEXT -1 130 "of \+ these equations. Now, these equations are recognized by the classifica tion methods of standard computer algebra system solvers." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "ode : = -y(t)+t*diff(y(t),t)-diff(y(t),t)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG,(-%\"yG6#%\"tG!\"\"*&F)\"\"\"-%%diffG6$F&F)F,F,*$)F-\"\" #F,F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dsolve(ode,y(t)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"yG6#%\"tG,$*$)F'\"\"#\"\"\"#F ,\"\"%/F$,&*&F'F,%$_C1GF,F,*$)F2F+F,!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 263 16 "gener al solution" }{TEXT -1 93 " consists of straight lines while the singu lar solution is a parabola. It turns out that the " }{TEXT 264 18 "sin gular solution " }{TEXT -1 6 "is an " }{TEXT 265 8 "envelope" }{TEXT -1 57 " of the family of straight lines of the general solution." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "plot([t^2/4,seq(-(c/2)^2+c*t/2,c=-3..3)], t=-4..4, y=-2..3);" }} {PARA 13 "" 1 "" {GLPLOT2D 394 213 213 {PLOTDATA 2 "6,-%'CURVESG6$7S7$ $!\"%\"\"!$\"\"%F*7$$!3nmmmmFiDQ!#<$\"3A!e4#)QZ)eOF07$$!35LLLo!)*Qn$F0 $\"3&*[e7a+++:v2*\\#F0$\"3.WY\\1rMh:F07$$!3BLLL8>1DBF0$\"31:$3_Ay9N\" F07$$!3lmmmw))yr@F0$\"3VlK?Jn;z6F07$$!3:+++S(R#**>F0$\"3mq\"))\\%))R#* **!#=7$$!30++++@)f#=F0$\"3ND5!QdEbL)Fdo7$$!3-+++gi,f;F0$\"3zo4Oxt$3)oF do7$$!3qmmm\"G&R2:F0$\"3ZBt(zL,1o&Fdo7$$!3WLLLtK5F8F0$\"3%37I_u2IS%Fdo 7$$!3fLLL$HsV<\"F0$\"39$[s$3d(yW$Fdo7$$!3+-++]&)4n**Fdo$\"3W.`jPjd$[#F do7$$!37PLLL\\[%R)Fdo$\"3'*4!*RKWoh6\"Fdo7$$!3Y******\\O3E]Fdo$\"3CH$*>9#z`J'!#>7$$!3NKLLL3z6LFdo$\"3 8sw$4j*)>u#Fbr7$$!3sLLL$)[`P(*** ***z-6j'Fdo$\"3,&>/'3\")G*4\"Fdo7$$\"3q\"******4#32$)Fdo$\"3q1NSD.>DqM8F0$\"3(\\3\\\"\\Hd`WFdo7$$\"3%)*******HSu]\" F0$\"3,C-;X1%4o&Fdo7$$\"3(HLL$ep'Rm\"F0$\"3$*4F1'4l>#pFdo7$$\"3()***** *R>4N=F0$\"3:3Cjqg!*=%)Fdo7$$\"3$emm;@2h*>F0$\"3))y\"p6+56'**Fdo7$$\"3 ^*****\\c9W;#F0$\"3EMgH-E:nEF0$\"3qtR@7\\Uy+++XhUkOF0$\"3$3*)RIu/qN$F07$ $\"3=+++:o$\"3p*****\\lfsu#F07$FC$\"3t++]s@%3 \\#F07$FH$\"3J++]U.6`AF07$FM$\"3')****\\-G&p+#F07$FR$\"3(*****\\AjP_F07$F_s $!3*)*****\\#p2WAF07$Fes$!3-****\\xgk3DF07$Fjs$!3Q****\\-V&*QFF07$F_t$ !3P*****\\\\$p()HF07$Fdt$!3e******>amWKF07$Fit$!3v)****\\Jig\\$F07$F^u $!31****\\PEd*)pF07$Fdy$!3M+++&o#RbsF07$Fiy$!3++++?`9$\\(F 07$F^z$!3H++]<#Rmu(F07$Fcz$!3E++]A_E*)zF07$F+$!3+++++++]#)F0-Fiz6&F[[l F_[lF\\[lF_[l-F$6$7S7$F($\"\"$F*7$F.$\"3nmmmmFiDGF07$F4$\"35LLLo!)*Qn# F07$F9$\"3nmmmwxE.DF07$F>$\"3YmmmOk]JBF07$FC$\"3_LLL[9cg@F07$FH$\"3smm mhN2-?F07$FM$\"3!******\\`oz$=F07$FR$\"3!omm;)3Do;F07$FW$\"3>+++:v2*\\ \"F07$Ffn$\"3BLLL8>1D8F07$F[o$\"3lmmmw))yr6F07$F`o$\"3a,+++uR#***Fdo7$ Ffo$\"3_++++5#)f#)Fdo7$F[p$\"3?++++E;!f'Fdo7$F`p$\"3)pmmm\"G&R2&Fdo7$F ep$\"3WMLLLF.rKFdo7$Fjp$\"3%fLLL$HsVF07$Fcu$!3H******H% =H<#F07$Fhu$!35mmm1>qMBF07$F]v$!3%)*******HSu]#F07$Fbv$!3(HLL$ep'Rm#F0 7$Fgv$!3()******R>4NGF07$F\\w$!3$emm;@2h*HF07$Faw$!3^*****\\c9W;$F07$F fw$!3Lmmmmd'*GLF07$F[x$!3j*****\\iN7]$F07$F`x$!3aLLLt>:nOF07$Fex$!36LL L.a#o$QF07$Fjx$!3ammm^Q40SF07$F_y$!3z******z]rfTF07$Fdy$!3gmmmc%GpL%F0 7$Fiy$!3/LLL8-V&\\%F07$F^z$!3>+++XhUkYF07$Fcz$!3=+++:o$\"3BLLL=Kv:9F07$FC$\"3 wmm;C2GI8F07$FH$\"3OLL$3yO5D\"F07$FM$\"3&*****\\nU)*o6F07$FR$\"3SLL$3W DT3\"F07$FW$\"3(4++]d(Q&***Fdo7$Ffn$\"3:mmmm&4`7*Fdo7$F[o$\"3BLLL$QW*e $)Fdo7$F`o$\"3x++++()>'\\(Fdo7$Ffo$\"3E++++0\"*HmFdo7$F[p$\"35++++83&z &Fdo7$F`p$\"3\\LLL3k(p.&Fdo7$Fep$\"3Anmmmj^NTFdo7$Fjp$\"3(zmmmYh=P$Fdo 7$F_q$\"3+,++v#\\N[#Fdo7$Fdq$\"3commmCC(p\"Fdo7$Fiq$\"3P\"*****\\FRX$) Fbr7$F^r$\"3$H(*****\\#=/8F]s7$Fdr$!3CQLLLe/T%)Fbr7$Fir$!39LLLeDBJ;Fdo 7$F_s$!3Immm;kD![#Fdo7$Fes$!3Rjmm\"f`@O$Fdo7$Fjs$!3%z****\\nZ)HTFdo7$F _t$!3bkmm;$y*e\\Fdo7$Fdt$!3f)******R^b\"eFdo7$Fit$!3&e*****\\5a`mFdo7$ F^u$!3'o****\\7RVY(Fdo7$Fcu$!3X'*****\\@fk$)Fdo7$Fhu$!3_ILLL&4N<*Fdo7$ F]v$!3#*******\\,s.5F07$Fbv$!3[mm;zM)>3\"F07$Fgv$!3$*******pfan6F07$F \\w$!3\"HLLeg`![7F07$Faw$!3w****\\#G2AL\"F07$Ffw$!3;LLL$)G[99F07$F[x$! 3\")****\\7yh+:F07$F`x$!3xmmm')fd$e\"F07$Fex$!3bmmm,FTo;F07$Fjx$!3FLL$ e#pa_F07$Fiy$!3_mmm1^r(*>F 07$F^z$!35++]sI@#3#F07$Fcz$!34++]2%)3j@F07$F+$!3+++++++]AF0-Fiz6&F[[lF _[lF_[lF\\[l-F$6$7S7$F(F_[l7$F.F_[l7$F4F_[l7$F9F_[l7$F>F_[l7$FCF_[l7$F HF_[l7$FMF_[l7$FRF_[l7$FWF_[l7$FfnF_[l7$F[oF_[l7$F`oF_[l7$FfoF_[l7$F[p F_[l7$F`pF_[l7$FepF_[l7$FjpF_[l7$F_qF_[l7$FdqF_[l7$FiqF_[l7$F^rF_[l7$F drF_[l7$FirF_[l7$F_sF_[l7$FesF_[l7$FjsF_[l7$F_tF_[l7$FdtF_[l7$FitF_[l7 $F^uF_[l7$FcuF_[l7$FhuF_[l7$F]vF_[l7$FbvF_[l7$FgvF_[l7$F\\wF_[l7$FawF_ [l7$FfwF_[l7$F[xF_[l7$F`xF_[l7$FexF_[l7$FjxF_[l7$F_yF_[l7$FdyF_[l7$Fiy F_[l7$F^zF_[l7$FczF_[l7$F+F_[l-Fiz6&F[[lF\\[lF_[lF\\[l-F$6$7S7$F(Fdgm7 $F.$!3cLLL$Q6G;#F07$F4$!3bmm;M!\\p3#F07$F9$!36LLL))Qj,?F07$F>$!3BLLL=K v:>F07$FC$!3wmm;C2GI=F07$FH$!3OLL$3yO5v\"F07$FM$!3&*****\\nU)*o;F07$FR $!3SLL$3WDTe\"F07$FW$!35++]d(Q&*\\\"F07$Ffn$!3immmc4`79F07$F[o$!3KLLLQ W*eL\"F07$F`o$!33+++q)>'\\7F07$Ffo$!3.+++]5*H;\"F07$F[p$!3,+++I\"3&z5F 07$F`p$!3NLL$3k(p.5F07$Fep$!3Anmmmj^N\"*Fdo7$Fjp$!3(zmmmYh=P)Fdo7$F_q$ !3+,++v#\\N[(Fdo7$Fdq$!3commmCC(p'Fdo7$Fiq$!39*****\\FRX$eFdo7$F^r$!3t *****\\#=/8]Fdo7$Fdr$!3=mmm;a*e:%Fdo7$Fir$!3'omm;Wn(oLFdo7$F_s$!3qLLL$ eV(>DFdo7$Fes$!3hOLL3k%yj\"Fdo7$Fjs$!3e?++]K_,()Fbr7$F_t$!3eWNLL$o@5%F ]s7$Fdt$\"3%f)******R^b\")Fbr7$Fit$\"3&e*****\\5a`;Fdo7$F^u$\"3'o**** \\7RVY#Fdo7$Fcu$\"3X'*****\\@fkLFdo7$Fhu$\"3_ILLL&4N<%Fdo7$F]v$\"3A*** ****\\,s.&Fdo7$Fbv$\"3%[mm;zM)>eFdo7$Fgv$\"3L*******pfan'Fdo7$F\\w$\"3 8HLLeg`![(Fdo7$Faw$\"3c(****\\#G2A$)Fdo7$Ffw$\"3jJLLL)G[9*Fdo7$F[x$\"3 \")****\\7yh+5F07$F`x$\"3xmmm')fd$3\"F07$Fex$\"3bmmm,FTo6F07$Fjx$\"3FL L$e#pa_7F07$F_y$\"3*)******Rv&)H8F07$Fdy$\"3ILLLGUY=9F07$Fiy$\"3_mmm1^ r(\\\"F07$F^z$\"35++]sI@#e\"F07$Fcz$\"34++]2%)3j;F07$F+Fd^m-Fiz6&F[[lF _[lF\\[lF\\[l-F$6$7S7$F(F\\^m7$F.$!37nmmmFiD[F07$F4$!35LLLo!)*Qn%F07$F 9$!3AmmmwxE.XF07$F>$!3YmmmOk]JVF07$FC$!3_LLL[9cgTF07$FH$!3GmmmhN2-SF07 $FM$!3!******\\`oz$QF07$FR$!3!omm;)3DoOF07$FW$!3>+++:v2*\\$F07$Ffn$!3B LLL8>1DLF07$F[o$!3lmmmw))yrJF07$F`o$!3:+++S(R#**HF07$Ffo$!30++++@)f#GF 07$F[p$!3-+++gi,fEF07$F`p$!3qmmm\"G&R2DF07$Fep$!3WLLLtK5FBF07$Fjp$!3fL LL$HsV<#F07$F_q$!3?+++b)4n*>F07$Fdq$!3rLLL$\\[%R=F07$Fiq$!3$)*****\\&y !pm\"F07$F^r$!3&******\\O3E]\"F07$Fdr$!3CLLL$3z6L\"F07$Fir$!3PLLL)[`P< \"F07$F_s$!3ummm;([R+\"F07$Fes$!3Atmm;Gpv#)Fdo7$Fjs$!37/++]YISnFdo7$F_ t$!3*3nmmOV?3&Fdo7$Fdt$!3\"G+++?(*)oLFdo7$Fit$!3I3+++z\"Hp\"Fdo7$F^u$! 3)*G1++]qM$Fdo7$F]v$\"3 W)*******HSu]Fdo7$Fbv$\"3oHLL$ep'RmFdo7$Fgv$\"3m)******R>4N)Fdo7$F\\w$ \"3Eemm;@2h**Fdo7$Faw$\"3^*****\\c9W;\"F07$Ffw$\"3Lmmmmd'*G8F07$F[x$\" 3j*****\\iN7]\"F07$F`x$\"3aLLLt>:n;F07$Fex$\"36LLL.a#o$=F07$Fjx$\"3amm m^Q40?F07$F_y$\"3z******z]rf@F07$Fdy$\"3gmmmc%GpL#F07$Fiy$\"3/LLL8-V& \\#F07$F^z$\"3>+++XhUkEF07$Fcz$\"3=+++:o$!3p*****\\lfsC(F07$FC$!3t++]s@%3*pF07$FH$!3J++]U.6`nF07$FM$!3')**** \\-G&p]'F07$FR$!3(*****\\AjP_iF07$FW$!3^++]sih)*fF07$Ffn$!3H+++qGfPdF0 7$F[o$!3_*****\\J$o2bF07$F`o$!3B+++5'f)[_F07$Ffo$!3j******\\J(*))\\F07 $F[p$!3D+++!RC&QZF07$F`p$!3g****\\AH46XF07$Fep$!3%*******4\\lSUF07$Fjp $!3&*******R%e:,%F07$F_q$!33++]#yk]u$F07$Fdq$!3N+++SF<4NF07$Fiq$!3u*** *\\#yh.D$F07$F^r$!3q****\\ZD\"R+$F07$Fdr$!3j*****\\ionu#F07$Fir$!31++] K-j5DF07$F_s$!36+++vI#fD#F07$Fes$!3)4++D#RN\"*>F07$Fjs$!3i++](pX5w\"F0 7$F_t$!3j+++0lI7:F07$Fdt$!3U+++!eM`D\"F07$Fit$!3D,++&oPR+\"F07$F^u$!3V 4++DE)pg(Fdo7$Fcu$!3l5++]NA1\\Fdo7$Fhu$!3V3+++9ZzCFdo7$F]v$\"3kw****** \\/;6Fbr7$Fbv$\"3_%****\\P/&fCFdo7$Fgv$\"3*z******4zj-&Fdo7$F\\w$\"3R( )***\\<3;W(Fdo7$Faw$\"3p#****\\Z=i'**Fdo7$Ffw$\"3\\******\\'[MC\"F07$F [x$\"3W****\\PM&=]\"F07$F`x$\"3v+++gzs]Ed*[#F07$Fdy$\"3M+++&o#RbFF07$Fiy$ \"3++++?`9$*HF07$F^z$\"3H++]<#RmC$F07$Fcz$\"3E++]A_E*[$F07$F+Fd[lFfdl- %+AXESLABELSG6$Q\"t6\"Q\"yFjfo-%%VIEWG6$;F(F+;$!\"#F*F\\el" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C urve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "A genera l way to determine if there are any singular solutions to a differenti al equation is provided by these command of the " }{TEXT 266 8 "diffal g " }{TEXT -1 70 "package. it will also determine the equation of the \+ singular solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "R := differential_ring(ranking=[y], derivat ions=[t], notation=diff):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "equations(Rosenfeld_Groebner([ode],R));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7#,(-%\"yG6#%\"tG\"\"\"*&F)F*-%%diffG6$F&F)F*!\"\"*$) F,\"\"#F*F*7#,&F&\"\"%*$)F)F2F*F/" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 20 "Arbitrary constants " }}{PARA 0 "" 0 "" {TEXT -1 55 "Consider t he following ordinary differential equations." }}{PARA 0 "" 0 "" {TEXT -1 19 "As unveiled by the " }{TEXT 267 28 "diffalg[Rosenfeld_Gro ebner] " }{TEXT -1 40 "command, it admits 2 singular solutions." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "ode := diff(y(t),t)^3-4*t*y(t)*diff(y(t),t)+8*y(t)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG,(*$)-%%diffG6$-%\"yG6#%\"tGF.\"\"$\"\" \"F0**\"\"%F0F.F0F+F0F(F0!\"\"*&\"\")F0)F+\"\"#F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "R := differential_ring(ranking=[y], deriv ations=[t],notation=diff):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "S:=equations(Rosenfeld_Groebner([ode],R));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG7$7#,(*$)-%%diffG6$-%\"yG6#%\"tGF0\"\"$\"\"\"F2** \"\"%F2F0F2F-F2F*F2!\"\"*&\"\")F2)F-\"\"#F2F27#,&*$F8F2\"#F*(F4F2F-F2) F0F1F2F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "The singular solution s consist of a cubic and the zero function." }}{PARA 0 "" 0 "" {TEXT -1 36 "The solutions under closed form are:" }}{PARA 0 "" 0 "" {TEXT -1 24 " " }{XPPEDIT 18 0 "y(t) = a*(t-a)^2;" "6 #/-%\"yG6#%\"tG*&%\"aG\"\"\"*$,&F'F*F)!\"\"\"\"#F*" }{TEXT -1 13 " \+ " }{XPPEDIT 18 0 "y(t)= 4/27*t^3" "6#/-%\"yG6#%\"tG*(\"\"%\" \"\"\"#F!\"\"F'\"\"$" }{TEXT -1 12 " " }{XPPEDIT 18 0 "y(t) =0" "6#/-%\"yG6#%\"tG\"\"!" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 91 "We see that, while the cubic cannot be obtained from the genera l solution, by specializing " }{XPPEDIT 18 0 "a = 0;" "6#/%\"aG\"\"!" }{TEXT -1 83 " in the closed form of the general solution we obtain th e second singular solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "Note that both the cubic and the zero function \+ are envelopes of the non-singular solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "d1:=plot(4*t^3/2 7,t=-5..5,y=-15..15, thickness=3): \nd2:= plot(\{ seq((a/6)^3*(t-(a/6) ^3)^2,a=-9..9)\},t=-5..5,y=-15..15): plots[display](\{d1,d2\});" }} {PARA 13 "" 1 "" {GLPLOT2D 332 233 233 {PLOTDATA 2 "68-%'CURVESG6$7S7$ $!\"&\"\"!$F*F*7$$!3XLLLe%G?y%!#f_PF/F+7$$!3K++vo1Y ZNF/F+7$$!3;LL3-OJNLF/F+7$$!3p***\\P*o%Q7$F/F+7$$!3Kmmm\"RFj!HF/F+7$$! 33LL$e4OZr#F/F+7$$!3v*****\\n\\!*\\#F/F+7$$!3%)*****\\ixCG#F/F+7$$!3\" ******\\KqP2#F/F+7$$!3:LL3-TC%)=F/F+7$$!3Zmmm\"4z)e;F/F+7$$!3Lmmmm`'zY \"F/F+7$$!3#****\\(=t)eC\"F/F+7$$!3\"ommmh5$\\5F/F+7$$!3S$***\\(=[jL)! #=F+7$$!3*f***\\iXg#G'FcoF+7$$!3ndmmT&Q(RTFcoF+7$$!3$\\mmTg=><#FcoF+7$ $!3uDMLLe*e$\\!#?F+7$$\"3zrm;zRQb@FcoF+7$$\"3_,+](=>Y2%FcoF+7$$\"3rumm \"zXu9'FcoF+7$$\"3$4+++]y))G)FcoF+7$$\"3H++]i_QQ5F/F+7$$\"3b++D\"y%3T7 F/F+7$$\"3+++]P![hY\"F/F+7$$\"3jKLL$Qx$o;F/F+7$$\"3Z+++v.I%)=F/F+7$$\" 3@mm\"zpe*z?F/F+7$$\"3;,++D\\'QH#F/F+7$$\"3&HL$e9S8&\\#F/F+7$$\"3s++D1 #=bq#F/F+7$$\"3\"HLL$3s?6HF/F+7$$\"3a***\\7`Wl7$F/F+7$$\"3fnmmm*RRL$F/ F+7$$\"3%zmmTvJga$F/F+7$$\"3]MLe9tOcPF/F+7$$\"32,++]Qk\\RF/F+7$$\"3![L L3dg6<%F/F+7$$\"3&ymmmw(GpVF/F+7$$\"3C++D\"oK0e%F/F+7$$\"36,+v=5s#y%F/ F+7$$\"\"&F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-F$6$7S7$F($!31apk_]Eb6F co7$F-$!3D'\\$R]`kc5Fco7$F1$!3e@La?*fTu*!#>7$F4$!3JL$e_@$>f))F\\v7$F7$ !3i^#\\d2-4,)F\\v7$F:$!3'oSGq8@!4sF\\v7$F=$!3**)*4p>LM.lF\\v7$F@$!3d!* 3L%*)\\4\"eF\\v7$FC$!3()R\"[;8fe8&F\\v7$FF$!3c8rK8dS/XF\\v7$FI$!3t+4Ve 'y!)*QF\\v7$FL$!3dY9&GX8.S$F\\v7$FO$!3uV9krjh!)GF\\v7$FR$!3!ou?)*oD@S# F\\v7$FU$!3D'y88M.@)>F\\v7$FX$!3&\\l+;EDcj\"F\\v7$Fen$!3`Ig[gs\"pE\"F \\v7$Fhn$!3cg4(fDkO\"**F`p7$F[o$!3\">(z#f(f'H8(F`p7$F^o$!3Y)G^;-&e_]F` p7$Fao$!3]OXsF6r\"=$F`p7$Feo$!3]7_)eHM0!=F`p7$Fho$!3zS;7P@`dx!#@7$F[p$ !3Q'*fFddaUV!#F7$Fbp$!3(3V'=J[;WAFiy7$Fep$!3o;:o 28,iyFiy7$Fhp$!3WK?gP*Qgx\"F`p7$F[q$!3vRw.:;W;KF`p7$F^q$!3>+FN$y\"[O]F `p7$Faq$!3R()GT*fzU=(F`p7$Fdq$!3e')=-QaZ,5F\\v7$Fgq$!3RR[eE8\"eH\"F\\v 7$Fjq$!3&>cV\"Q!y=l\"F\\v7$F]r$!3w!Q%e)[4=,#F\\v7$F`r$!3@c4D#3peW#F\\v 7$Fcr$!3I2T]C9(H*GF\\v7$Ffr$!3*>Pk\\E(*e\"ea,d))F\\v7$Fat$!3!3#*pgl+Kt*F\\v7$Fdt$!33=GIS@0h5Fco7$Fgt$! 37j%[nu^&f6Fco-Fjt6&F\\uF+F]uF+-F$6$7S7$F($!3bE_k+JfA\"*Fco7$F-$!3rM0i &*z')Q$)Fco7$F1$!3a8I\\5fe&o(Fco7$F4$!3,1)=`iKF)pFco7$F7$!3*p&G(>t`#4j Fco7$F:$!3\\z$3J]yGn&Fco7$F=$!3#e*4'G(>48^Fco7$F@$!3X)y@*3:5kXFco7$FC$ !3C\\)>Cf@\"HSFco7$FF$!3F&=u!>_.HNFco7$FI$!3qf)HnB&>\\IFco7$FL$!3q\"p@ YJ#ebEFco7$FO$!3$fF=sI-]C#Fco7$FR$!3=CdGz'3u'=Fco7$FU$!35E0&3#4SO:Fco7 $FX$!3[EWNE\"oPE\"Fco7$Fen$!30!HMGA<@u*F\\v7$Fhn$!3Lq')48.a$e(F\\v7$F[ o$!3s-n\"fg#H7aF\\v7$F^o$!3$*zp4Pq<&z$F\\v7$Fao$!3tk&=v%>D]BF\\v7$Feo$ !3mzN(yf6YH\"F\\v7$Fho$!3[%*QtWFFi_F`p7$F[p$!3B*\\;&*>l??\"F`p7$F^p$!3 4su.\"p/m\"Q!#A7$Fbp$!3jGBP;PviBF`p7$Fep$!3Dd()RbEvx5Fco7$Fjq$!34+M#H4OsO\"Fc o7$F]r$!3@+rfE\"y)f;Fco7$F`r$!3izqnC2E7?Fco7$Fcr$!3#[u')4;uZP#Fco7$Ffr $!3))yjK37y&y#Fco7$Fir$!3!)*)Q(3k7$>KFco7$F\\s$!3ks,:W'enq$Fco7$F_s$!3 0D#=N])p3UFco7$Fbs$!3!)emS#Gc\\v%Fco7$Fes$!3L>;WJ()fH`Fco7$Fhs$!3\\U`w !fFl)eFco7$F[t$!3PH#R9*=')elFco7$F^t$!3xXJQh$**4>(Fco7$Fat$!3m%3cnb=q* yFco7$Fdt$!3(>sBQyGPg)Fco7$Fgt$!3a)zTJ_TpR*Fco-Fjt6&F\\uF]uF]uF+-F$6$7 S7$F($\"37j%[nu^&f6Fco7$F-$\"3U81.p^ug5Fco7$F1$\"36'*4\\8@`$y*F\\v7$F4 $\"3$Q%Ra3ot'*))F\\v7$F7$\"3)pCF#[\\gY!)F\\v7$F:$\"3kvMKP?*GC(F\\v7$F= $\"39IPEZd^NlF\\v7$F@$\"3`A!zF\\v7$FX$\"3dhqv<'zoQ&Rv$f?Fiy7$Fep$\"3S*[UpQyE^(Fiy7$Fhp$\"3nlIP)\\MLs\"F`p 7$F[q$\"3E\"\\*)=%yPXJF`p7$F^q$\"3(4lbP)pXZ\\F`p7$Faq$\"3(=v(=$ewy2(F` p7$Fdq$\"3dmILBdj \"F\\v7$F]r$\"3o(*3A&>xR*>F\\v7$F`r$\"3TWOw()G?ECF\\v7$Fcr$\"3AH&y>nz: (GF\\v7$Ffr$\"3Ke=42=AxLF\\v7$Fir$\"30k%)Gl0?6RF\\v7$F\\s$\"3$H)fH-.?7 XF\\v7$F_s$\"3lA*)RfPiJ^F\\v7$Fbs$\"3*RPz%HME1eF\\v7$Fes$\"3K#\\ ))F\\v7$Fat$\"3#Q&p@r*HRp*F\\v7$Fdt$\"3Gqn!zs^p0\"Fco7$Fgt$\"31apk_]Eb 6Fco-Fjt6&F\\uF+F+F]u-F$6$7S7$F($\"3a)zTJ_TpR*Fco7$F-$\"3-GgR)Qc7g)Fco 7$F1$\"3/E\"e`%zcPzFco7$F4$\"3-+nMA;,BsFco7$F7$\"3JF?$e6_x`'Fco7$F:$\" 3%>\"o*\\E_'*)eFco7$F=$\"3-ukK4b**=`Fco7$F@$\"3YF)*ec&\\(eZFco7$FC$\"3 ULCZ(>H@@%Fco7$FF$\"3)4TeDrR/q$Fco7$FI$\"3qDBw.Xm3KFco7$FL$\"3Nn*)pF!R X!GFco7$FO$\"36SOVuV7#Q#Fco7$FR$\"3Y=iW1vk#*>Fco7$FU$\"3\"pJ2V/)=];Fco 7$FX$\"30GhLR;F\\v7$Fho$\"3)y*Gk@UtLvF`p7$F[p$\"3kbY?&3!z$R#F`p7$F^p$\" 3$[J-?;:\\_'Ficl7$Fbp$\"3!H\"[DF85!=\"F`p7$Fep$\"3t))*oE'>-#3&F`p7$Fhp $\"3Ctd(>J'4O7F\\v7$F[q$\"3z0_\"=&[KABF\\v7$F^q$\"3i07D\"H(p8PF\\v7$Fa q$\"3,X#QU'RPp`F\\v7$Fdq$\"3?KY!4#)*GkvF\\v7$Fgq$\"3%)z[kk/cc)*F\\v7$F jq$\"3.6_%e7XQE\"Fco7$F]r$\"3D332\\9vX:Fco7$F`r$\"3/'Ge*zqR')=Fco7$Fcr $\"3a\\I#[%p'yB#Fco7$Ffr$\"3D!HUzFItj#Fco7$Fir$\"3'*40%)>cdfIFco7$F\\s $\"3=Gu9Kh?NNFco7$F_s$\"3@qvSpiwDSFco7$Fbs$\"3Q%)4$)QmQgXFco7$Fes$\"3n %*o-U!)[B^Fco7$Fhs$\"3w&=J)y=\")pcFco7$F[t$\"3)\\i@pi\"**HjFco7$F^t$\" 3*3VR,#zD^pFco7$Fat$\"3C)4(pthoXwFco7$Fdt$\"3LqmZ!R-8M)Fco7$Fgt$\"3bE_ k+JfA\"*Fco-Fjt6&F\\uF]uF+F]u-F$6$7S7$F($!#;F*7$F-$!3-@V'f#RPI9Fj[n7$F 1$!3UA`CzS^!H\"Fj[n7$F4$!3fGp0b8#=9\"Fj[n7$F7$!3bWd21?L,5Fj[n7$F:$!3k% >'*RG)eeb*['F/7$FC$!3cKa2?'*o`aF/7 $FF$!3nA:4Gcs5XF/7$FI$!3ZN'=U7%3MOF/7$FL$!3(=u`$y)>.%HF/7$FO$!3$[g<\"G *\\rC#F/7$FR$!3CScie)[Zk\"F/7$FU$!3W01&3r#)H:\"F/7$FX$!3@,;p?j()=yFco7 $Fen$!3q9\\Nul@TVFco7$Fhn$!3[uY*R%e\"**=#Fco7$F[o$!3A1k?_t0YgF\\v7$F^o $!36ep/;p`JCF`p7$Fao$!3?dXB`ttnFF\\v7$Feo$!3\\%euQ)G!>Q\"Fco7$Fho$!33B F+OkEMMFco7$F[p$!3^(>zR$e)y7'Fco7$F^p$!3]1,,9d_,**Fco7$Fbp$!3+=)4ofLvZ \"F/7$Fep$!3'>VJF0\\4)>F/7$Fhp$!3%y8[f0+ug#F/7$F[q$!3zAw(ywI[M$F/7$F^q $!3=&>x$yW,bTF/7$Faq$!3%GO]n*4YA]F/7$Fdq$!3*45lG9')=3'F/7$Fgq$!3\"G%[) )fyB?rF/7$Fjq$!3yV^A`')=>$)F/7$F]r$!3oDD(3eXh[*F/7$F`r$!3IEXT9Y&\\3\"F j[n7$Fcr$!3aS(*)z<'f@7Fj[n7$Ffr$!3IA]o8F/7$FI$\"3AzNV>#='[6F/7$FL$\"3aUtuOE,35F/7$FO $\"3Z0X3re/2')Fco7$FR$\"3+]XfVc$\\C(Fco7$FU$\"3Gcdix'QK/'Fco7$FX$\"3AC e>FtKY]Fco7$Fen$\"3zjo5n2yxRFco7$Fhn$\"3KL$\\UKB><$Fco7$F[o$\"3$\\&p)3 0l\"\\BFco7$F^o$\"3/f+_!ycPs\"Fco7$Fao$\"350)\\Y*es[6Fco7$Feo$\"3.qW(o V;D4(F\\v7$Fho$\"3ay3W$>g6j$F\\v7$F[p$\"3,UFpm3pj9F\\v7$F^p$\"3Zf8DGrT 5@F`p7$Fbp$\"3X&fkr=]Y-\"F`p7$Fep$\"3'3Y\"*H%>4t**F`p7$Fhp$\"3#G?*G5>7 )*HF\\v7$F[q$\"3(HG?sJEK>'F\\v7$F^q$\"3+V->(zSG/\"Fco7$Faq$\"3,FBn[l0d :Fco7$Fdq$\"3yEE6tvM[AFco7$Fgq$\"30c5B%='G)o*zeFco7$Fcr$\"35Lh&)e!>>-(Fco7 $Ffr$\"3X^Y)fwUQK)Fco7$Fir$\"3(4y5(f#)o.(*Fco7$F\\s$\"3L(fMmpeh7\"F/7$ F_s$\"3HM)3P'>;(G\"F/7$Fbs$\"3m$3xGGKHY\"F/7$Fes$\"3@A!y;d`$[;F/7$Fhs$ \"3w`yTsv[G=F/7$F[t$\"3M#z?1#pUY?F/7$F^t$\"3[TXyItu^AF/7$Fat$\"39_3@M9 Z\"[#F/7$Fdt$\"3j$>jgj&z6FF/7$Fgt$\"3++++DJqqHF/Fit-F$6$7S7$F($\"3f\"= qUlM8J)F/7$F-$\"3wXXDj%)HTwF/7$F1$\"3#>;vm!*473(F/7$F4$\"37.=KUjzwkF/7 $F7$\"3gQwB?Ig&*eF/7$F:$\"3/190)>%HW`F/7$F=$\"3&Q'3Li@Ld[F/7$F@$\"3$fr 3:.?wP%F/7$FC$\"3tE:*[]Ix!RF/7$FF$\"3%[!*oX?!*eY$F/7$FI$\"3X2rY.90RIF/ 7$FL$\"3L*)=areJ'o#F/7$FO$\"3Bwi^SrC:BF/7$FR$\"3t.F(Rd#Rq>F/7$FU$\"3-7 7*QBgVm\"F/7$FX$\"334Np>n\")39F/7$Fen$\"3F@#p60bE8\"F/7$Fhn$\"3()>&RIX vDA*Fco7$F[o$\"3F\\60+Y!p/(Fco7$F^o$\"3wb\"fl[7\\O&Fco7$Fao$\"3a5nSZh% Hy$Fco7$Feo$\"3s.Vg9hvKDFco7$Fho$\"3A=%H*ykw%\\\"Fco7$F[p$\"3i?_sCqW7y F\\v7$F^p$\"3*Gp<6H<')o#F\\v7$Fbp$\"3vmY4*['RK>F`p7$Fep$\"3'f,l72p:m$F `p7$Fhp$\"3@8+hi-s/IF\\v7$F[q$\"3ca\"3I%fb/%)F\\v7$F^q$\"3V!G&)=&>pJ;F co7$Faq$\"39-'*H[c\"[k#Fco7$Fdq$\"3)>ftu8s\\0%Fco7$Fgq$\"3i:#*>hL4ybFc o7$Fjq$\"3%H&eBcC(=Z(Fco7$F]r$\"38lP\\\\J_E%*Fco7$F`r$\"3Xb._6`I#=\"F/ 7$Fcr$\"30UkO.#fDV\"F/7$Ffr$\"3ALZ/qu!)>=;?\"HJF/7$Fes$\"3n Pm-cjGZNF/7$Fhs$\"3idcTi8laRF/7$F[t$\"3abVl8Cw[WF/7$F^t$\"3,)\\&R`hL: \\F/7$Fat$\"3\")p)e\">TUQaF/7$Fdt$\"3tTfxjQ&Q'fF/7$Fgt$\"3`J@H]Z]blF/F i^l-F$6$7S7$F($\"3DDk)e!z.,=Fj[n7$F-$\"3NV([_$o/j;Fj[n7$F1$\"3&3v9\"Ra XZ:Fj[n7$F4$\"33h`hyVVA9Fj[n7$F7$\"3krn\")4B!>I\"Fj[n7$F:$\"3$>6O:TSs= \"Fj[n7$F=$\"3*e`!o6_l&3\"Fj[n7$F@$\"3s\")Hzi]c_)*F/7$FC$\"3E%o%p+'pa' ))F/7$FF$\"39;r\"f;!QLzF/7$FI$\"3Z/ic:@hGqF/7$FL$\"3Y(QYt\">0xiF/7$FO$ \"3<(*)Ho%*3=[&F/7$FR$\"3&>P\">RgZPZF/7$FU$\"3^/59o%Q:2%F/7$FX$\"3KKp: ?9[5NF/7$Fen$\"3E79=e-W(*GF/7$Fhn$\"3?X1?)[0TU#F/7$F[o$\"3sIya\\7eE>F/ 7$F^o$\"3K!=Y/-9Q`\"F/7$Fao$\"3/Q%4oLSV:\"F/7$Feo$\"3HIcnX%Q.V)Fco7$Fh o$\"37pgpRof-dFco7$F[p$\"3$ye'[erzlOFco7$F^p$\"3K/p?:%o7(>Fco7$Fbp$\"3 @0&oObnCj(F\\v7$Fep$\"3av%3.@wpp\"F\\v7$Fhp$\"3mRJ!>6Tq^(Fiy7$F[q$\"3[ ;(3xTGAi$F\\v7$F^q$\"3OA%GA\"Fco7$Faq$\"3&[/F*H\\0RDFco7$Fdq$\"3XI rS1TidXFco7$Fgq$\"3FF_uYDYroFco7$Fjq$\"3)*Qf)H.$[k)*Fco7$F]r$\"3qK_>aI E/8F/7$F`r$\"3XrpxKyT-CJ6Fj[nFbhl-F$6$7S7$F($\"#OF*7$F-$\"3eawH4`=VLFj[n7$ F1$\"3(*))>T8JYFJFj[n7$F4$\"3qh-RV_X$*GFj[n7$F7$\"3yx!4WA&3nEFj[n7$F:$ \"3'fGm:%[%4X#Fj[n7$F=$\"37F2IEIreAFj[n7$F@$\"3&)>GQ`)Rz1#Fj[n7$FC$\"3 zw3%GS%\\z=Fj[n7$FF$\"3'>:4.K61q\"Fj[n7$FI$\"3.I&)3p$Rf_\"Fj[n7$FL$\"3 Z2(ohUE*z8Fj[n7$FO$\"3Mg<\"G'[LC7Fj[n7$FR$\"3#Rcie$fYx5Fj[n7$FU$\"3V/1 &3,k![%*F/7$FX$\"3R$\\-//k)=$)F/7$Fen$\"3.d@5C!Q'pqF/7$Fhn$\"3tKh1^I&3 4'F/7$F[o$\"38k?_[)4S/&F/7$F^o$\"3[9Fe.Sn*>%F/7$Fao$\"3.VB`[m@iLF/7$Fe o$\"3,duQ8@B^EF/7$Fho$\"3\\NpEg?K**>F/7$F[p$\"3%eek]-c:[\"F/7$F^p$\"3N WVtah*)45F/7$Fbp$\"3n\"\\J90+Q:'Fco7$Fep$\"339VJxP,6NFco7$Fhp$\"3^\"o9 GRx$GMZ\"F`p7$Faq$\"3Dhg. vr=7eF\\v7$Fdq$\"3*R,^'G*RH<#Fco7$Fgq$\"3yE^^lKGnWFco7$Fjq$\"3^99DK:() >yFco7$F]r$\"31he?*y5j;\"F/7$F`r$\"3`e_9Wk3u;F/7$Fcr$\"3/uSc@dUNAF/7$F fr$\"36?-&=N#z3HF/7$Fir$\"3(>H&=$*Hr_OF/7$F\\s$\"3#*z#*QV;>AXF/7$F_s$ \"3f0S+odFZaF/7$Fbs$\"3\"***\\A$pxA['F/7$Fes$\"3vUL!Hxgvf(F/7$Fhs$\"3> GG%=%))R+()F/7$F[t$\"3:+3gOfi05Fj[n7$F^t$\"3Em4Y0+@N6Fj[n7$Fat$\"3jJ* \\\"G9-#G\"Fj[n7$Fdt$\"3d/$p0$y*3V\"Fj[n7$Fgt$\"#;F*F[bm-F$6$7S7$F($\" 3bN_i2(e>*oFj[n7$F-$\"3I`K#*zSWVkFj[n7$F1$\"3\">I2GFqa1'Fj[n7$F4$\"3;G 8BFm/acFj[n7$F7$\"3SmPh,w[a_Fj[n7$F:$\"3+D&G(3MOr[Fj[n7$F=$\"3lG[l?G6H XFj[n7$F@$\"3!RT&3tt'y=%Fj[n7$FC$\"3gS1S'=3!\\QFj[n7$FF$\"3C`/4J2YDNFj [n7$FI$\"3E=a#>9qu?$Fj[n7$FL$\"3H#[$)*R1$)RHFj[n7$FO$\"3**>EC'H\"[_EFj [n7$FR$\"3cDt9+y\")yBFj[n7$FU$\"3zLdS)*o=H@Fj[n7$FX$\"3Y/CX&Q$[9>Fj[n7 $Fen$\"30+z03z-u;Fj[n7$Fhn$\"3IjghZ1&H[\"Fj[n7$F[o$\"3?LD![fY_F\"Fj[n7 $F^o$\"30B_/&*=Y/6Fj[n7$Fao$\"3$fBBi[I?J*F/7$Feo$\"3KG&QCK7&*z(F/7$Fho $\"3%*oi=%=gTO'F/7$F[p$\"3])ziEg5X<&F/7$F^p$\"3Q)H4E%4=HSF/7$Fbp$\"3:A hYZj+\"*HF/7$Fep$\"3aM8'ytdH@#F/7$Fhp$\"3x&>X!3b//:F/7$F[q$\"3L!z&4DOw \\\"*Fco7$F^q$\"3Og=iQU@'z%Fco7$Faq$\"3pEC!e132\">Fco7$Fdq$\"3Ve'3/cej N#F\\v7$Fgq$\"3>kR$e\"H&o-\"F\\v7$Fjq$\"3Gc*e0O$[%R\"Fco7$F]r$\"3epwBB +#Q%QFco7$F`r$\"3C]O'e,\"y7zFco7$Fcr$\"3kO$f_\"*GoI\"F/7$Ffr$\"3$pTf:9 aK)>F/7$Fir$\"3_ahrxR[!y#F/7$F\\s$\"3b9OU8(y!fPF/7$F_s$\"3#z3WD)H!3%[F /7$Fbs$\"3nix!Q-.$)3'F/7$Fes$\"3#*[S!\\,mlY(F/7$Fhs$\"3c]stk:#p&))F/7$ F[t$\"3[V0nuLjf5Fj[n7$F^t$\"3d3Q+a7TG7Fj[n7$Fat$\"3.fjOEg4A9Fj[n7$Fdt$ \"3)\\O')*o5v?;Fj[n7$Fgt$\"3B559kfq[=Fj[nFc[n-F$6$7S7$F($\"3hj\">;6TwG \"!#:7$F-$\"3$)Qx2Veg77Fadq7$F1$\"38))fA,\"Fadq7$F:$\"3mj(=J$3/r%*Fj[n7$F=$\"3>**H'\\wwm)))F j[n7$F@$\"3tm^0,(37I)Fj[n7$FC$\"3YjC4[.q;xFj[n7$FF$\"3-!QH&[.IbrFj[n7$ FI$\"3n%RK'\\+&**f'Fj[n7$FL$\"3Q>_@dmPHhFj[n7$FO$\"3^\"G\\&=[W?cFj[n7$ FR$\"3Ae6`.NhJ^Fj[n7$FU$\"3h$f&4,Jd\"o%Fj[n7$FX$\"3YyxDOQy!H%Fj[n7$Fen $\"3K)Ri;.h#[QFj[n7$Fhn$\"3=smg\"HCA\\$Fj[n7$F[o$\"3cw+%z54)*4$Fj[n7$F ^o$\"39pCx(zi>x#Fj[n7$Fao$\"3z#GM>3RLV#Fj[n7$Feo$\"3<][))3lQJ@Fj[n7$Fh o$\"3<#R8m(okP=Fj[n7$F[p$\"3vk+NPj2(e\"Fj[n7$F^p$\"3%f&)HB*=QP8Fj[n7$F bp$\"3Pi$)pqUj+6Fj[n7$Fep$\"3*3$ycn(fI8*F/7$Fhp$\"3YBl36v+1tF/7$F[q$\" 3T\"H&*p2*RKcF/7$F^q$\"3?>6phRZ0UF/7$Faq$\"3R*f!=5+!H-$F/7$Fdq$\"3M:95 Yv0Q>F/7$Fgq$\"3()Re(4m/\"o6F/7$Fjq$\"3B/hY4GL+cFco7$F]r$\"3'\\&*HDBX \"**>Fco7$F`r$\"3i(ogC\\(R(Q\"F\\v7$Fcr$\"3:NIpw1r*o$F\\v7$Ffr$\"3BXy) *=l\\iEFco7$Fir$\"3OPG]^'RM$pFco7$F\\s$\"3$Rp!\\kiPb8F/7$F_s$\"3v^3+m \"43?#F/7$Fbs$\"3G\\D;#yxiF$F/7$Fes$\"3PX`XF/7$Fhs$\"3n#)Q4P>&>\" fF/7$F[t$\"3fW'\\S_Wno(F/7$F^t$\"3Tv+ms&>7Z*F/7$Fat$\"3nIqa_E)y:\"Fj[n 7$Fdt$\"3*zayQy@%z8Fj[n7$Fgt$\"3-/@tI(*4R;Fj[nF[en-F$6$7S7$F($\"3+++v$ 4YsO#Fadq7$F-$\"3S?X'GdFcC#Fadq7$F1$\"3nbhiCrTU@Fadq7$F4$\"392GqdnCH?F adq7$F7$\"3B?K-9vU=>Fadq7$F:$\"3WV5whVA6=Fadq7$F=$\"3xa[WPme9:Fadq7$FF$\"3\\N'3&>;VD9Fadq7$FI$ \"3>TRzS(3;L\"Fadq7$FL$\"3r>_%H*[h^7Fadq7$FO$\"3'oSs4^DX;\"Fadq7$FR$\" 3+G9OT!Q-3\"Fadq7$FU$\"3RqFMfq+-5Fadq7$FX$\"3O#=)Gx88N$*Fj[n7$Fen$\"3 \">giWJHAb)Fj[n7$Fhn$\"38>Bn?3$e\"zFj[n7$F[o$\"3d!HZ#pB]1sFj[n7$F^o$\" 39:H0'*>S1mFj[n7$Fao$\"3=e#)=v[+yfFj[n7$Feo$\"3iZSp>o!)3aFj[n7$Fho$\"3 )p-,y'*e_%[Fj[n7$F[p$\"3vm(*=(zY]N%Fj[n7$F^p$\"37a1tZ()ebQFj[n7$Fbp$\" 3jX=nU=Fj[n7$Faq$\"3TRVFH!Qo`\"Fj[n7$Fdq $\"3>%yuu3a(H7Fj[n7$Fgq$\"38m)[0b#*)H)*F/7$Fjq$\"3t%4;/jv)*\\(F/7$F]r$ \"3$eA$f7/KgcF/7$F`r$\"3-zR3X!z[%RF/7$Fcr$\"3`$p*)R,/Gh#F/7$Ffr$\"3Vo \"\\&=[p7:F/7$Fir$\"3ylh1%o^(fsFco7$F\\s$\"3$yls$[:R$3#Fco7$F_s$\"34& \\.&)Q,+p&F`p7$Fbs$\"3?U!)*\\1.D()*F\\v7$Fes$\"3l5qM;Zj3\\Fco7$Fhs$\"3 0Iq!f\\xW6\"F/7$F[t$\"3l2^5MoJR@F/7$F^t$\"3l\"fK*\\DbOLF/7$Fat$\"3[C6x G!=\\!\\F/7$Fdt$\"3iv!y[#[;)o'F/7$Fgt$\"3++++v$4@\"*)F/Fit-F$6$7S7$F($ !3++++DJqqHF/7$F-$!36=erd#*)4r#F/7$F1$!3$*[l6Fsn%\\#F/7$F4$!3;_M],Y:iA F/7$F7$!3UA2(Q#pdR?F/7$F:$!39q\"yFH*\\H=F/7$F=$!3,rCN@z#\\k\"F/7$F@$!3 ](eU'GY:k9F/7$FC$!3Ce^'zlLNT7\"F/7$FI$!3!p7WXdF(p' *Fco7$FL$!3g$fxvI;MQ)Fco7$FO$!3N0&4U#)R^/(Fco7$FR$!3#)\\&pz7(Q=eFco7$F U$!3*ov+XAKru%Fco7$FX$!3GTPRjZnoQFco7$Fen$!3_(>:[L\")4%HFco7$Fhn$!3(om #3q\\WaAFco7$F[o$!3)\\?omZ&[q:Fco7$F^o$!3H#R.^kPz1\"Fco7$Fao$!33b0JH8/ xiF\\v7$Feo$!3%=(>J&e))e;$F\\v7$Fho$!3n<#**[gBQ/\"F\\v7$F[p$!3pkd)3R!=F`p7$Fbp$!3NHcp0+e\\9F\\v7$Fep$!3'ok'['=YRa$F \\v7$Fhp$!3'[(33IbFSoF\\v7$F[q$!3HG?(z`xt8\"Fco7$F^q$!3KVFD'eJ=p\"Fco7 $Faq$!3NxN&p`MFL#Fco7$Fdq$!3zE,b'4!pkJFco7$Fgq$!3%*)Q9)[ID?SFco7$Fjq$! 39G*=i**4m/&Fco7$F]r$!3x\\zRuNIxgFco7$F`r$!3\\w#>B:\"F/7$F\\s$!3UAx$) Hxc@8F/7$F_s$!3!4]D;@Lb\\\"F/7$Fbs$!3H+D^n#fXo\"F/7$Fes$!3t![R)Gl7$)=F /7$Fhs$!30aG/8.Mv?F/7$F[t$!3WvyHcW72BF/7$F^t$!3637?;y#[_#F/7$Fat$!30x* )yEZvnFF/7$Fdt$!3\")o]t*p:2,$F/7$Fgt$!3++++DJ?$G$F/Fi^l-F$6$7S7$F($!3+ +++v$4@\"*)F/7$F-$!3B&eROl&e\"o'F/7$F1$!3N)po&Rlt,]F/7$F4$!35l>Jx\"GES $F/7$F7$!3o[2aw)[I5#F/7$F:$!3Do\\**Q\\e=6F/7$F=$!3t^f)oFH>\"[Fco7$F@$! 3<&Gnk_:Q+\"Fco7$FC$!3Kg+:c$fcJ&F`p7$FF$!3@e?zV&y)G@Fco7$FI$!33+l\"e'H K8uFco7$FL$!38dd6GfKr9F/7$FO$!3LqJ0!G+'*e#F/7$FR$!3?)G!3ZtTGSF/7$FU$!3 wWXY,ba9dF/7$FX$!3+\")*4<(=W+vF/7$Fen$!3$o`j0`9'R**F/7$Fhn$!3s?tz=OTF7 Fj[n7$F[o$!32.E>t#G*H:Fj[n7$F^o$!3T:a'))\\![D=Fj[n7$Fao$!3pt&eABc(z@Fj [n7$Feo$!3(o)\\!f6&HYDFj[n7$Fho$!35cTv%f!4fHFj[n7$F[p$!3AHj;$QmaO$Fj[n 7$F^p$!3_G+Ms&*4LQFj[n7$Fbp$!3(3Z$e?&Q5N%Fj[n7$Fep$!3I_HHJ'='G[Fj[n7$F hp$!3Pg!f)feMs`Fj[n7$F[q$!3tZ**=cr_kfFj[n7$F^q$!35;w*yp9Qd'Fj[n7$Faq$! 3VbY%QcI:>(Fj[n7$Fdq$!38fTL$3\"*)4zFj[n7$Fgq$!3h$QnyqLXe)Fj[n7$Fjq$!3o h`jY6LN$*Fj[n7$F]r$!3O8U'eQ%G/5Fadq7$F`r$!3GlR!p*4f%3\"Fadq7$Fcr$!30GW `vM(H;\"Fadq7$Ffr$!3W[\"34oGyC\"Fadq7$Fir$!3:T4)f`yOL\"Fadq7$F\\s$!3w` \"*fV_hE9Fadq7$F_s$!3\\&QEtH$3>:Fadq7$Fbs$!3x(o)Hozk;;Fadq7$Fes$!3\"*y LU#\\.kr\"Fadq7$Fhs$!3v?`6a7q5=Fadq7$F[t$!3F*)pU.#y=#>Fadq7$F^t$!3]Eo= kA7C?Fadq7$Fat$!3fUA=$Qag8#Fadq7$Fdt$!3:]nT7*3gC#Fadq7$Fgt$!3+++v$4YsO #FadqFbhl-F$6$7S7$F($!3-/@tI(*4R;Fj[n7$F-$!3JcKpi)H'y8Fj[n7$F1$!3+%*[o .:Kq6Fj[n7$F4$!3@_vrRkGk&*F/7$F7$!3>#H=r8$**GwF/7$F:$!3&p/'oha()>fF/7$ F=$!3=w-P8\"z'GXF/7$F@$!30KJs&oYUG$F/7$FC$!3sXkQ^%)32AF/7$FF$!3)oEbT&H sX8F/7$FI$!3B][E)>())3oFco7$FL$!3,FY+d*o4\"GFco7$FO$!3O,<*zSX\\#RF\\v7 $FR$!3VW6LPI9J=F\\v7$FU$!3-*4Z0V__3#Fco7$FX$!3<3wD*fH;g&Fco7$Fen$!3\\+ 3&)3&G**>\"F/7$Fhn$!3h7&f%f]FI>F/7$F[o$!38@FUvNC(*HF/7$F^o$!3W$*3i)pnn 8%F/7$Fao$!3k-JBRFw(f&F/7$Feo$!3QgUuK$[R>(F/7$Fho$!3^XYrcDcs!*F/7$F[p$ !3)e>sT#e%*)4\"Fj[n7$F^p$!3*RySEl)GE8Fj[n7$Fbp$!3k'*3G*o[]e\"Fj[n7$Fep $!3u&*))\\o71H=Fj[n7$Fhp$!3vwC$G+;A6#Fj[n7$F[q$!3'QM9D3MhU#Fj[n7$F^q$! 3gRgs-\"zUv#Fj[n7$Faq$!3o'H]&H7e\"4$Fj[n7$Fdq$!3I@P_h\"=*)[$Fj[n7$Fgq$ !3H_D4*pDk'QFj[n7$Fjq$!3O&H<'Rt*3H%Fj[n7$F]r$!3v1A!Q;?Yp%Fj[n7$F`r$!3i =U<&Gin:&Fj[n7$Fcr$!3C?;u%G49h&Fj[n7$Ffr$!3&e6AP2vr5'Fj[n7$Fir$!3'HBU! yF;7mFj[n7$F\\s$!3iS93=&GB;(Fj[n7$F_s$!3'yYsdA&)Hr(Fj[n7$Fbs$!3N$)e2z. ?(H)Fj[n7$Fes$!3W#[t0.Rw*))Fj[n7$Fhs$!3KMC`a2(yY*Fj[n7$F[t$!3v-Yq[+K95 Fadq7$F^t$!3\")[31RJpw5Fadq7$Fat$!3J0VE1dCX6Fadq7$Fdt$!3%[PPboSG@\"Fad q7$Fgt$!3hj\">;6TwG\"FadqF[bm-F$6$7S7$F($!3B559kfq[=Fj[n7$F-$!3*yST:X[ +i\"Fj[n7$F1$!3zKkqW6PL9Fj[n7$F4$!3];Gx?/3P7Fj[n7$F7$!3CO,Xd.3a5Fj[n7$ F:$!3b^89*fe['))F/7$F=$!3$\\31:*))eSuF/7$F@$!3$o\\-2B#>(4'F/7$FC$!3u\\ B$e\"HU[[F/7$FF$!314ekw%3fu$F/7$FI$!3W[;GOU,gFF/7$FL$!3c%fSN(e5;?F/7$F O$!3px>L.Z8=8F/7$FR$!3Ll.GI\"\\&fwFco7$FU$!3S]\\KAAtZPFco7$FX$!3\"3hD2 !Q&RR\"Fco7$Fen$!3d%=DfE?g)zF`p7$Fhn$!3'QCl(f]d'G#F\\v7$F[o$!3*QX!48n; e=Fco7$F^o$!3)\\W'4zlT2YFco7$Fao$!3O&=Mx7$oN!*Fco7$Feo$!3F1wA$yfDY\"F/ 7$Fho$!3Q=FnK1h)=#F/7$F[p$!3N0_%3r.Q)HF/7$F^p$!3-wK?-]RzRF/7$Fbp$!3Ev \\q!eN];&F/7$Fep$!3v.R1Fj[n7$F]r$!3'pz_@o*QO@ Fj[n7$F`r$!3nMjRDf$GR#Fj[n7$Fcr$!3*Rd2W=,uk#Fj[n7$Ffr$!3\\3$*RPyCFHFj[ n7$Fir$!3LUs#e9RW@$Fj[n7$F\\s$!3z$z7$G()\\HNFj[n7$F_s$!3R%3b\\ngo%QFj[ n7$Fbs$!3A@u\"p*oFj[nFc[n-F$6$7S7$F($!3`J@H]Z]blF/7$ F-$!3[(R\"\\\\F,ifF/7$F1$!3zU%QN)[_oaF/7$F4$!3`'HV8w5!R\\F/7$F7$!3yi*Q NO8KV%F/7$F:$!3PUa'>7Up&RF/7$F=$!3_0=&*G&\\&RNF/7$F@$!3IqKVED(=8$F/7$F C$!3W#*\\bK=[OFF/7$FF$!3@l)piW,*oBF/7$FI$!3PM$fW4[%=?F/7$FL$!3&4LZy!H* Ht\"F/7$FO$!3))\\zL5fmP9F/7$FR$!3WVb9!3k)o6F/7$FU$!3ZavnPkAh$*Fco7$FX$ !3:62F:HMruFco7$Fen$!36^wAz98,bFco7$Fhn$!3=VYWv/enSFco7$F[o$!3IyXbx@xr EFco7$F^o$!3:)=\\zc!4!o\"Fco7$Fao$!3fmc*))fU]b)F\\v7$Feo$!3zg,j97>lKF \\v7$Fho$!3AwcwqL6.TF`p7$F[p$!3W5-@jv2a=F`p7$F^p$!3VVf(4U&G:DF\\v7$Fbp $!3(\\ch#R\\@ixF\\v7$Fep$!3F,(fSL$[n9Fco7$Fhp$!3Yw%zq&eCfCFco7$F[q$!3= j*[CuF7v$Fco7$F^q$!3@![!QVx9y_Fco7$Faq$!30A7)G>$3.qFco7$Fdq$!3YhgFt_e. #*Fco7$Fgq$!3V&Qvir'oV6F/7$Fjq$!3)o*R#\\V*))39F/7$F]r$!3?Vw6^I1t;F/7$F `r$!37I1sPE$y)>F/7$Fcr$!3;(39oQl(3BF/7$Ffr$!38&*G]CM*)pEF/7$Fir$!3Y>*e Vi>$[IF/7$F\\s$!3DmTp,)f8Z$F/7$F_s$!3\"ov!46\\x/RF/7$Fbs$!3phNq!)eOuVF /7$Fes$!3:m3GoZRm[F/7$Fhs$!39dUz=>jT`F/7$F[t$!3+@qc1@`8fF/7$F^t$!3:e_N x8o\\kF/7$Fat$!3Yqjtq8&p/(F/7$Fdt$!3Fkq>\"y#QVwF/7$Fgt$!3f\"=qUlM8J)F/ F[en-F$6$7S7$F($!3Qcc__>CJ6Fj[n7$F-$!3'Q?*>t,XA5Fj[n7$F1$!3j)[tIC\\EK* F/7$F4$!3%oLx_Ld\"e$)F/7$F7$!3F.5[t#\\/W(F/7$F:$!3iKoDWf2!e'F/7$F=$!3q `hc3.hHeF/7$F@$!3!3^A:Y:/5&F/7$FC$!3#oflH\\5vR%F/7$FF$!3Y0C@\\,q[PF/7$ FI$!3kDkSj%>`8$F/7$FL$!3>LK[9MTSEF/7$FO$!3%=\"[qFA5M@F/7$FR$!3Hy$oa[() )z;F/7$FU$!3i9_nFco7$F hn$!3tiYmO'4jd%Fco7$F[o$!3deLj1o+wDFco7$F^o$!3&*)GDs5g;G\"Fco7$Fao$!3$ HTm]/))4w$F\\v7$Feo$!3$oc$)=1A7U\"F`p7$Fho$!3p))>OLfOq:F\\v7$F[p$!3)3( )pGuDJc(F\\v7$F^p$!3ak$*4av90>Fco7$Fbp$!3c%yN(*o\"e]OFco7$Fep$!36K\"yv ]C!GcFco7$Fhp$!3c[zjqceU#)Fco7$F[q$!39-phRPfY6F/7$F^q$!37!GNcr'H8:F/7$ Faq$!3et/^@EX;>F/7$Fdq$!3(oPJdX-)>CF/7$Fgq$!3`4^u%4\"4AHF/7$Fjq$!3K]4[ N=k5NF/7$F]r$!3Bk%RA*)e04%F/7$F`r$!3'pP>9sg_x%F/7$Fcr$!3S@9?(RpyY&F/7$ Ffr$!3_GV;iQ'>C'F/7$Fir$!3K&GY%G\"4$[qF/7$F\\s$!3:'yUPsW\\%zF/7$F_s$!3 \">9'\\pzCf))F/7$Fbs$!3UEL(QCUd%)*F/7$Fes$!3skVc`'[v3\"Fj[n7$Fhs$!38L& zX&eo'=\"Fj[n7$F[t$!3nP3'\\hDcI\"Fj[n7$F^t$!3eN)fLE=oT\"Fj[n7$Fat$!3lM J?>tPS:Fj[n7$Fdt$!37AhlllZj;Fj[n7$Fgt$!3DDk)e!z.,=Fj[nFit-F$6%7S7$F($! 3!>&=&=&=&=&=Fj[n7$F-$!3_f_&p&f1?;Fj[n7$F1$!3dpviUN&[V\"Fj[n7$F4$!3AR# R%fT2W7Fj[n7$F7$!38[`\"yV7*p5Fj[n7$F:$!3]Yb#>Q>_8*F/7$F=$!3*ReE,!4rGyF /7$F@$!3zI#3kdoPh'F/7$FC$!3>4pflLv'\\&F/7$FF$!3HYuEI(>h^%F/7$FI$!3+k^[ 3s)oj$F/7$FL$!3\"Hl&f2g+kHF/7$FO$!3'[+&[Eg<7BF/7$FR$!3t)>Fv_M;w\"F/7$F U$!3!yx#*HHH7K\"F/7$FX$!3Sq[>xtx5**Fco7$Fen$!3S:l_$>;Iw'Fco7$Fhn$!3!\\ oU_TYko%Fco7$F[o$!3ED$)['4_]'GFco7$F^o$!3FK<,RUi6*y)\\GM:y\"!#D7$Fbp$\"3cq5(*f1W$[\"F`p7$Fep$\"3bR>a2n?- 5F\\v7$Fhp$\"3Opgwe\"e&=&=&=&=&=Fj[nFit-%*THICKNESSG6#\"\"$-%+AXESLABELSG6$Q\"t6\"Q\"yF [iv-%%VIEWG6$;F(Fgt;$FadqF*$\"#:F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1 8" "Curve 19" "Curve 20" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 130 "We could have a-priori, i.e. without pre vious knowledge of the closed form solution, unveiled the fact that th e zero function was " }{TEXT 268 9 "embedded " }{TEXT -1 79 "in the ge neral solution. This can be done through the following command of the \+ " }{TEXT 269 8 "diffalg " }{TEXT -1 8 "package." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "M := equati ons(essential_components" }{TEXT -1 0 "" }{MPLTEXT 1 0 10 "(ode, R)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG7$7#,(*$)-%%diffG6$-%\"yG6#% \"tGF0\"\"$\"\"\"F2*(F0F2F-F2F*F2!\"%*$)F-\"\"#F2\"\")7#,&*$)F0F1F2F4F -\"#F" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "We have here a minimal \+ description of the solution set of the ordinary differential equation \+ under consideration." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Note that dsolve now uses diffalg to compute th e singular solutions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "ds olve(ode, y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/-%\"yG6#%\"tG\"\" !/F$,$*$)F'\"\"$\"\"\"#\"\"%\"#F/F$,(*&*$)F'\"\"#F.F.%$_C1G!\"\"#F.F0* &#F.\"\")F.*&F'F.*$)F8F7F.F9F.F9*&#F.\"#kF.*$)F8F-F.F9F." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "Envelope?" }}{PARA 0 "" 0 "" {TEXT -1 61 " Consider the following ordinary differential equations. With " }{TEXT 270 28 "diffalg[Rosenfeld_Groebner] " }{TEXT -1 46 "we find out that t here is a singular solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 " \+ " }{XPPEDIT 18 0 "diff(y(t),t)^2 -y(t)^3 = 0 " "6#/,&*$ -%%diffG6$-%\"yG6#%\"tGF,\"\"#\"\"\"*$-F*6#F,\"\"$!\"\"\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "p \+ := diff(y(t),t)^2 - y(t)^3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "equations(Rosenfeld_Groebner([p],R)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7#,&*$)-%%diffG6$-%\"yG6#%\"tGF.\"\"#\"\"\"F0*$)F+\" \"$F0!\"\"7#F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "Plotting the solutions, we realize that the singula r solution is not an envelope of the non-singular solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "p lot(\{seq(4/(t-3*_C1)^2, _C1=-3..3)\},t=-15..15, 0..1/2);" }}{PARA 13 "" 1 "" {GLPLOT2D 325 236 236 {PLOTDATA 2 "6+-%'CURVESG6$7gq7$$!#:\"\" !$\"3066666666!#=7$$!3&*****\\P&3YV\"!#;$\"3h)e;@L\\&*R\"F-7$$!3++]iv< rx8F1$\"3W'p#HzFy_F17$$!3 U\\(oHp:5V*FR$\"3F-ZG6%*FR$\"3ZJc0]]xlBF17$$!3O]i: &ol8R*FR$\"37Kc<'*o_6EF17$$!3&****\\7oS:P*FR$\"3H#*fO&zew*GF17$$!3\"yo HzA[6N*FR$\"3m5=MKk)RC$F17$$!3mv$4Yxb2L*FR$\"32(=a(pxKcOF17$$!3_j!*G@L O5$*FR$\"3vI+M#\\\"f_TF17$$!3g\\(oz'3(**G*FR$\"3DC$RdZF17$$!3oN%[Y Ty&p#*FR$\"3DML6qO9/bF17$$!3aB\"G8'f=\\#*FR$\"3)H(eCOK)=W'F17$$!3S6y+3 NzG#*FR$\"3g.PCnSRTwF17$$!3D*\\(oa5S3#*FR$\"36'ox\"o8,5#*F17$$!36(=n8g 3!)=*FR$\"3(p&zCl-jJ6F)7$$!3(\\(o/[hhn\"*FR$\"3(4^7z5KPU\"F)7$$!3$GcEZ pBs9*FR$\"3Rg?c,(fa%=F)7$$!3o]iST7$o7*FR$\"3lmhP?!4m[#F)7$$!3aQf3)yQk5 *FR$\"3:2*=qc%pINF)7$$!3SEcwMj/'3*FR$\"3x6+`td]-aF)7$$!3D9`W\")Qll!*FR $\"3!4*=l/?\")z#*F)7$$!3M+]7G9EX!*FR$\"3eX(=1%fb_>!#97$$!3U'o/[(*o[-*F R$\"35mFU)*fhnkF^w7$$!3FuV[@lZ/!*FR$\"3,(y0cA$3'*>!#77$$!38iS;oS3%)*)F R$\"30R<([))\\!z:!#87$$!3**\\P%[h\"pj*)FR$\"3$>h4FT8U.$F^w7$$!3&yVB:;* HV*)FR$\"3'y6V]ttTC\"F^w7$$!3qDJ?3n!H#*)FR$\"32Qb+))FR$\"3,Ni->Ga05F)7$$!3:P4' \\`f,y)FR$\"3VbW,e>Yw#)F17$$!3,D1k\"3n(f()FR$\"32_=Qp\\)4$pF17$$!3(GJ? $GYPR()FR$\"3PMiSKtz))eF17$$!3s+++v@)*=()FR$\"3=va%eM\\^1&F17$$!38Ec, \"\\?5q)FR$\"3*Hgx(*>O[Z%F17$$!3b^7.2)eIo)FR$\"3.$)fDhC,#)RF17$$!3'p(o /Br4l')FR$\"3'ej([MG=xT#F17$$!3Z+]7.()Gv&)FR$\"3h/kJ6YampNjO:&)=F17$$!3M+v=n>W.&)FR$\"3I(*G(4;eAi\"F17$$!3R\\(=#*f=vY) FR$\"3UR?h+Mv59F17$$!3@++DJ_fJ%)FR$\"3Nf')y!>o!Q7F17$$!3&***\\Pf\"FR7$$!3'3++](GVZoFR$\"3m!yv&H()pK')F-7$$ !3I+++D>PMlFR$\"3G#4A8I\"ozlF-7$$!3u*****\\(4J@iFR$\"3KdD)3Y+1=&F-7$$! 3*)***\\iIKFl&FR$\"3AG!zZY*3qNF-7$$!3')*****\\FPm(\\FR$\"3/YATn/0rCF-7 $$!3w++++h*QS%FR$\"3yFeNFac$*=F-7$$!33,+Dc>mPPFR$\"3.J0xt*\\WW\"F-7$$! 3'3+++&=$z9$FR$\"3`3s#Gp&*z;\"F-7$$!3!)***\\iX/4]#FR$\"3FQwg444q%*!#>7 $$!3p***\\(o8y%)=FR$\"3uT4CKg-,zFcdl7$$!3'*****\\i:#>C\"FR$\"3mh=:-s'e k'Fcdl7$$!3#f)**\\7ev:lF-$\"3)o^5ih%>RdFcdl7$$!3!*Q****\\(o2[\"Fcdl$\" 3dhutf;ca\\Fcdl7$$\"3i(***\\P>:mkF-$\"3r$*HC4AV)H%Fcdl7$$\"3o)**\\iv&Q A7FR$\"3g.#y%R[&y#QFcdl7$$\"3_,+]PPBW=FR$\"3=+1=\"fN9S$Fcdl7$$\"3R**** **\\Nm'[#FR$\"3Q*evq.-;.$Fcdl7$$\"3)3++vyb^6$FR$\"3C/T;@EADFFcdl7$$\"3 ')***\\PMaKs$FR$\"3k:H:9y%4Z#Fcdl7$$\"3B)***\\7TW)R%FR$\"3![-ob.&=GAFc dl7$$\"3*y*****\\@80]FR$\"3RUTB&*3KR?Fcdl7$$\"3k*****\\7,Hl&FR$\"3EBkf j&**H'=Fcdl7$$\"3j)**\\P4w)RiFR$\"3\"))zX:n`As\"Fcdl7$$\"3\\.++vZf\")o FR$\"3mN**G(G&)ee\"Fcdl7$$\"3t***\\P/-a[(FR$\"3]$=-)G5%=Z\"Fcdl7$$\"3G ,+v=Yb;\")FR$\"3J1(yo[(Hl8Fcdl7$$\"3$y****\\i@Ot)FR$\"3beVkyb$>F\"Fcdl 7$$\"3R++v$fL'z$*FR$\"3,S\"\\hI%4%=\"Fcdl7$$\"35+++!*>=+5F1$\"3()=AJ25 #y5\"Fcdl7$$\"3?++DE&4Q1\"F1$\"3R0YvMq>P5Fcdl7$$\"3<+]P%>5p7\"F1$\"3(f VQ>ULit*!#?7$$\"3K+++bJ*[=\"F1$\"3?%z]%z^@-#*Fb[m7$$\"3E++Dr\"[8D\"F1$ \"3N\\]=uS[U')Fb[m7$$\"3#)******Hjy58F1$\"3!y1!QKc,%=)Fb[m7$$\"3D+]P/) fTP\"F1$\"3]n5:?pCMxFb[m7$$\"3:+]i0j\"[V\"F1$\"3)p/$zO)3wL(Fb[m7$$\"#: F*$\"31WWWWWWWpFb[m-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*Fh]m-F$6$7hq7$F($ \"3Lr#Q\\gr#Q\\Fcdl7$F/$\"3j,j!*e>TUdFcdl7$F5$\"3f`q%)psO8mFcdl7$F:$\" 3$\\?K(\\TJ_yFcdl7$F?$\"3ia.)H@Wu[*Fcdl7$FD$\"3'**)oF63)z;\"F-7$FI$\"3 +4$\\#ox&pW\"F-7$FT$\"37rn%z:,g&=F-7$Fhn$\"3R&*[;.0f#\\#F-7$Fjq$\"3Si_ gw'p)=NF-7$F]\\l$\"3S&=J5&Hi5aF-7$Fc`l$\"3D*=Hf=O,q)F-7$Fh`l$\"3kRyjTN o17FR7$F]al$\"3s;[*)f\\b%y\"FR7$$!3w****\\()*>ZL(FR$\"3K=buKtKXAFR7$Fb al$\"3%z%4!G^W1\"HFR7$$!3z++]7>')4qFR$\"31\"o$\\GmDARFR7$Fgal$\"3$RXQC OD*pbFR7$$!3e++++C!4p'FR$\"3b\">5^oj'z$)FR7$F\\bl$\"3f*)*)p3!*y+9F17$$ !3B++D1$R_\\'FR$\"3*RL)e$=44j\"F17$$!3;++](o1hX'FR$\"3L.8=***oF#>F17$$ !34++voS(pT'FR$\"3z8[OPTg+BF17$$!3-+++]9%yP'FR$\"3+lprCO#=!GF17$$!3V+] iS^FejFR$\"3,+&ydX4i6$F17$$!3&****\\7$)3(QjFR$\"3r#Q7ftQm[$F17$$!3Z** \\(=_U\">jFR$\"3KS,,r*os#RF17$$!3))****\\7id*H'FR$\"33Bb5ox-dWF17$$!3H +]7.*4+G'FR$\"3y_aTM*z;5&F17$$!3\")***\\PfV/E'FR$\"3@!4EJV?q*eF17$$!3L **\\P%Gx3C'FR$\"3jj\\x)o_R*oF17$Fabl$\"3`4]5t2%o;)F17$$!3.i:g\"*Ga.iFR $\"3P(G_#ot!\\l*F17$$!3JCJ?3[x&='FR$\"3&\\'*z;x2!f6F)7$$!3[(o/[s1!ohFR $\"30BT:2,7<9F)7$$!3w\\iST'Q-:'FR$\"39_s?jX8s6 j&yv.%zAF)7$$!3Av$4YZ-Z6'FR$\"3c\\h8OeGSIF)7$$!3]P4@\"RMp4'FR$\"3[\\[T db+dUF)7$$!3y*\\7yIm\"zgFR$\"3\"z)p')G$HBQ'F)7$$!31iSTC#)RhgFR$\"3QFSa $e!3h5F^w7$$!3NCc,T,jVgFR$\"3Dxt#zp$H,@F^w7$$!3_(=5z wp&>F^w7$$!3#)**\\iS;-PfFR$\"3M*fj$eK]35F^w7$$!35ilAdND>fFR$\"3/w*yrf! )\\8'F)7$$!3QC\"GQZ&[,fFR$\"3l!zqIIR:7%F)7$$!3b(oH/R())> #eHF)7$$!3%)\\7.2$\\f'eFR$\"3ZRi04J)fA#F)7$$!377GjB7=[eFR$\"3IZZagwVN< F)7$$!3HvVBSJTIeFR$\"3KU(Go%\\$3R\"F)7$$!3dPf$o0XE\"eFR$\"3O.Nbv-aR6F) 7$$!3')*\\PM(p([z&FR$\"3X**QR)**Hn]*F17$$!39i!R+*)3rx&FR$\"3k@.EjtX^!) F17$$!3UC1k13MfdFR$\"3\\&o*=uKX1pF17$$!3f(=UKss:u&FR$\"3g1Hr(f*R*)fF17 $$!3))\\P%)RY!Qs&FR$\"3;s5U#)>dV_F17$$!3;7`Wcl.1dFR$\"3\")f=XnC&)GYF17 $$!3Lvo/t%o#)o&FR$\"30scs7'>i6%F17$$!3hP%['*Q+0n&FR$\"3!>6@MBfUo$F17$F fbl$\"3'pwSi>\"*oJ$F17$$!3*[(ozmjZ5cFR$\"3LnFx[]GOEF17$$!3*)\\PMF/AobF R$\"3F.880g`X@F17$$!3*[i!*y[kf_&FR$\"3(Rl$)Gtu+y\"F17$$!3*)*\\P%[&3P[& FR$\"3(ol)H+zh+:F17$$!3))\\7`pm>*R&FR$\"3wK$)zx<936F17$$!3))**\\i!z%o9 `FR$\"3U%=\"*f%\\&o^)FR7$$!3()*\\7G.hc9&FR$\"3p%*)RlxR-[&FR7$F[cl$\"3> lsTq(\\%>QFR7$$!3`++D\")>XL[FR$\"3%=pUyLt$RHFR7$$!3J++](om-p%FR$\"3+'= @p\"e\"=L#FR7$$!34++v$R\"3ZXFR$\"31$G,fFh[*=FR7$F`cl$\"3t$H*\\bu8q:FR7 $$!3#4+D\"G!z22%FR$\"3KV-?hR%*pE$F-7$Fedl$\"3$Q'p1Y)o>O#F-7$Fjdl$\"3%Rd 3xEQow\"F-7$F_el$\"33eE#4YD$)R\"F-7$Fdel$\"3CdaQ4eh;6F-7$Fiel$\"3[8l\\ 7UQa!*Fcdl7$F^fl$\"38IoS!y\"HowFcdl7$Fcfl$\"3jY(yAE#o+lFcdl7$Fhfl$\"3- d\\&z4YPb&Fcdl7$F]gl$\"3(\\r.6/&G9[Fcdl7$Fbgl$\"3H$\\5^OQ4B%Fcdl7$Fggl $\"3G.R4V;L*p$Fcdl7$F\\hl$\"3U@W?$f-FI$Fcdl7$Fahl$\"3,3*GouAd%HFcdl7$F fhl$\"3#z??uLn*pEFcdl7$F[il$\"3(pttlUv0T#Fcdl7$F`il$\"3#Q$yHp;a*>#Fcdl 7$Feil$\"3p$)f#*z]D2?Fcdl7$Fjil$\"3eA6N,?kU=Fcdl7$F_jl$\"33p!38.&4\"p \"Fcdl7$Fdjl$\"3IWsT6Y9i:Fcdl7$Fijl$\"3Pg!4K')\\\\W\"Fcdl7$F^[m$\"3)** H'e*>$GT8Fcdl7$Fd[m$\"3ETgtDWbb7Fcdl7$Fi[m$\"3*RH`9vMq;\"Fcdl7$F^\\m$ \"3RV&o@#*eb4\"Fcdl7$Fc\\m$\"3K^#z3v\\j-\"Fcdl7$Fh\\m$\"3'*ocuP1>2JFcdl7$F5$\"3'>-EaKLRW$Fcdl7$F:$\"3G(*4&= k;C*QFcdl7$F?$\"3o&\\jCHK&QWFcdl7$FD$\"3PY\"QLF-7$Fgal$\"3'R&4\")zC?-FF-7$Fabl$\"3nhnl%Ga6Jn)[$FR 7$$!3+,v=#\\?U!RFR$\"3FW#yX&*zA*[FR7$Fecl$\"3`Vb3J'z4N(FR7$$!3.,voH%H- f$FR$\"3W*3)[D>?[6F17$$!3(4+DJ!pzUMFR$\"3fmb4:r4S?F17$$!3s]P%)R13pLFR$ \"3ruRj(=:k$HF17$$!3#4]ilPk`H$FR$\"3?(yDroY]e%F17$$!3O)=U2JNpF$FR$\"3P W>LZ&)e:_F17$$!3Cw=#\\C1&eKFR$\"30Xe/))3u&)fF17$$!3oj:5zr2SKFR$\"3J=_x 9-)*RpF17$$!36^7G8\"[;A$FR$\"3`_\"R2H:?9)F17$$!3bQ4YZ!>K?$FR$\"3OlqeiM q&o*F17$$!3*fiS;)**y%=$FR$\"3e%HF'3WRr6F)7$$!3V8.#e\"4OmJFR$\"3K6X$)[z HX9F)7$Fjcl$\"3BwX,BM$y#=F)7$$!3b8`%RC7x7$FR$\"3bAmo*R?CX#F)7$$!3!ei!* yj#\\2JFR$\"3/(o=Uf/=Y$F)7$$!3/Qf$=.ts3$FR$\"3(fPCkX*p^_F)7$$!3t]7yDM0 nIFR$\"3W*p$)\\xwk*))F)7$$!3Ujls>Q$o/$FR$\"34`yvIskB=F^w7$$!3mv=n8UhEI FR$\"3,Rk')oS>ZcF^w7$$!3\"z=P+\"G4#F_x7$$!3H8y]&Rbf'HFR$\"3bO2Gwl9^MF^w7$$!3`DJX*yNd%HFR $\"3l?q?.nTe8F^w7$$!3yP%)R$=;b#HFR$\"3=Q4\")\\Y,5sF)7$$!3Y]PMxlH0HFR$ \"3U%=--:O*fWF)7$$!3:j!*Grp2&)GFR$\"3KURXFviGIF)7$$!3SvVBlt&['GFR$\"3y 4i\"y*o:!>#F)7$$!3k(oz\"fxjWGFR$\"3:z3xQv/GFR$\"3Lg2AdbMV5F)7$$!3EDc,T*yRy#FR$\"3=2))Q< nrr&)F17$$!3^P4'\\LfPw#FR$\"3\"fP_c0>s;(F17$$!3?]i!*G(RNu#FR$\"3J88XPs i\"3'F17$$!38voz;05.FFR$\"3[A/BpcvPXF17$$!31+vo/8miEFR$\"3X\"HN+%*=]^$ F17$$!3$*\\(o/)Gy\"e#FR$\"3sUc&fY[pG#F17$F_dl$\"3w'4dVE/eg\"F17$$!3x** \\P%otoM#FR$\"3[\"z(GBp.x$*FR7$$!3u****\\7H%G>#FR$\"3`f&R]]`'RhFR7$$!3 r**\\iS@\")Q?FR$\"3hR)pWwc&HVFR7$Fedl$\"3?SsDDr<;KFR7$$!3$)**\\il9Nj:F R$\"3v$zUI)H-Q>FR7$Fjdl$\"3x*)H'>mYTH\"FR7$F_el$\"3R2u`9'=GD(F-7$Fdel$ \"3'>O'3mgk)[%F-7$Fiel$\"3ZfS@Fe,3IF-7$F^fl$\"3P!4xI^$fVAF-7$Fcfl$\"36 *fU/E]Xq\"F-7$Fhfl$\"3sf*3z;](G8F-7$F]gl$\"3)e$f=#*zlp5F-7$Fbgl$\"3N6p H-o8\\))Fcdl7$Fggl$\"3lLxMFXn2tFcdl7$F\\hl$\"3)QC'Fcdl7$Fahl$\" 3U&[b-K,CM&Fcdl7$Ffhl$\"3!e'*[SF&>&o%Fcdl7$F[il$\"3GUhs9NV'4%Fcdl7$F`i l$\"3-b]&=?F#QOFcdl7$Feil$\"34I4zxw#oB$Fcdl7$Fjil$\"3PKf+NCL0HFcdl7$F_ jl$\"3p>x(>XB+h#Fcdl7$Fdjl$\"3!ydv*e8?mBFcdl7$Fijl$\"3UJO_$*[c]@Fcdl7$ F^[m$\"3F^hA!RmX'>Fcdl7$Fd[m$\"3;,e`5\\89=Fcdl7$Fi[m$\"3(Hf2/')R?m\"Fc dl7$F^\\m$\"3e)**[U3W;a\"Fcdl7$Fc\\m$\"3K,]^d'QrU\"Fcdl7$Fh\\m$\"3IJ8= Yd3H8Fcdl7$F]]m$\"3$ycM7!zcM7Fcdl-Fb]m6&Fd]mFe]mFe]mFh]m-F$6$7co7$F($ \"3yxxxxxxxFcdl7$F5$\"3xE`MB>Q2@Fcdl7$F:$\"3J:? Blcmxm$Hh :$Fcdl7$FT$\"31t#Q*)y(oJNFcdl7$Fhn$\"3.R#Hoe^_*RFcdl7$Fjq$\"3=sQSO;ZaX Fcdl7$F]\\l$\"3uT\\;;#GL$F-$\"3([?Y-liIg$F17$$!3!4\\ilZcf`#F-$\"3^H7\"Hr*z>iF17$$!3!> *\\P4m**RFy^XXOo)F)7$$\"3_aPMF54#4\"F-$\"3U<-B%zRQN$F)7 $$\"3*R+D\"G.[0:F-$\"3'f#*Qo@e[w\"F)7$$\"3Y`i!*G'p)=>F-$\"3.*R8-@[j3\" F)7$$\"3#H](oH*eAL#F-$\"3d.vNv$4PN(F17$$\"3R_(o/B[cu#F-$\"3HvvL3h.1`F1 7$$\"3'=+]7`P!fJF-$\"3-58>l&4#3SF17$$\"3u**\\PMZf7[F-$\"3p)yi/2Oqs\"F1 7$Fiel$\"3qki8PF$oc*FR7$$\"3?#*****\\Z+X$*F-$\"32qp#\\(RP!e%FR7$F^fl$ \"3/zpRU&ppn#FR7$Fcfl$\"3Sl(=U7%F-7$Fbgl$\"3fq%QM$oX&)GF-7$Fggl$\"3:RS_zwdn?F-7$F\\hl$\"3y** )oTZ?nf\"F-7$Fahl$\"3\"H>HP))[u)QrFcdl7$Feil$\"3^y1J_uyrgFcdl7$Fjil$ \"3ASH_(R.TC&Fcdl7$F_jl$\"3Vm#f3n:ma%Fcdl7$Fdjl$\"3>,zMxWa)*RFcdl7$Fij l$\"3U&\\R9jMX`$Fcdl7$F^[m$\"3^&))fjP)y\\JFcdl7$Fd[m$\"3#H-6l4g!\\GFcd l7$Fi[m$\"3g?#H9\"o[aDFcdl7$F^\\m$\"3$[;FH!32GBFcdl7$Fc\\m$\"3if\"[pX! H=@Fcdl7$Fh\\m$\"3/oAYod(H%>Fcdl7$F]]mF\\`p-Fb]m6&Fd]mFh]mFh]mFe]m-F$6 $7gp7$F(Fd_p7$F/$\"3T6e*f:/%H8Fcdl7$F5$\"3\"=,0[6-6U\"Fcdl7$F:$\"3\">' [!)*eLg`\"Fcdl7$F?$\"3,4:84^Sm;Fcdl7$FD$\"3coPYN'fL\"=Fcdl7$FI$\"3&=e1 K&))on>Fcdl7$FT$\"3+%>!*Rf8#\\@Fcdl7$Fhn$\"3$eZ=U9-ZO#Fcdl7$Fjq$\"3b^F s\"HRMh#Fcdl7$F]\\l$\"3i^Cs*p&f7HFcdl7$Fc`l$\"3oieJIOy?KFcdl7$F]al$\"3 sQw2S*)3IOFcdl7$Fgal$\"3uH1MJ[!\\7%Fcdl7$Fabl$\"3(e4&*G_zSq%Fcdl7$Ffbl $\"38osl(y4EM&Fcdl7$F[cl$\"3>t!R+zkmG'Fcdl7$F`cl$\"3H\\8pBi\"pH(Fcdl7$ Fecl$\"3;_M@8=L6))Fcdl7$Fjcl$\"3.FCB+KGe5F-7$F_dl$\"3s!)[wc#z=K\"F-7$F edl$\"3Jd\"*f@%pjn\"F-7$Fjdl$\"3JsM,Wd(HA#F-7$F_el$\"3**4#Ro0\\)**HF-7 $Fdel$\"3\"Qc))4`#*3S%F-7$Fiel$\"3cbO$*GjFAsF-7$F^fl$\"3Hu.[wy&eE\"FR7 $$\"35+](ou4L`\"FR$\"3TLwyDVWf=FR7$Fcfl$\"3-)*z.#**pW*HFR7$$\"3*4+D1>T [+#FR$\"3i[x)pf7!RSFR7$$\"3Y++vV'[a;#FR$\"3+k5g&*))>VdFR7$$\"3#***\\(o 4cgK#FR$\"3%)>ONIco1))FR7$Fhfl$\"3_!)*)oCT%z^\"F17$$\"3e*\\(oz]AlDFR$ \"3L-8jP[2;@F17$$\"3w**\\P4myVEFR$\"3yhoQ_JF17$$\"3j\\(=UPnIo#FR$\" 3_Y*)GWxA#)RF17$$\"3&**\\i!R\"[Bs#FR$\"3?f,Y=Iq)=&F17$$\"3*[P%[@&))>u# FR$\"3#GXJ$z&>(3gF17$$\"3F]i!R!*G;w#FR$\"3uzvPd\"y'RqF17$$\"3?D\"GjGp7 y#FR$\"3rFi>hylg$)F17$$\"39++vo'44!GFR$\"37%Q=$G\"f\"45F)7$$\"32v=<^+b ?GFR$\"3Zb\\-7z9U7F)7$$\"3,]PfL/>SGFR$\"3'Hy#*p0Eic\"F)7$$\"3RDc,;3$)f GFR$\"3m6$)\\vF*e.#F)7$$\"3K+vV)>r%zGFR$\"3]25\">5dMv#F)7$$\"3Ev$f3e6 \"**GFR$\"3Iy(3)=D')HRF)7$$\"3k]7Gj>v=HFR$\"3rRmOHWYfgF)7$$\"3dDJqXBRQ HFR$\"3))e$*QU(yQ0\"F^w7$$\"3^+]7GF.eHFR$\"36'ozU!=6rAF^w7$$\"3Xvoa5Jn xHFR$\"3Pfp\\-LBC!)F^w7$$\"3Q](oH\\8t*HFR$\"30^;'pSBAa&Fiw7$$\"3wD1RvQ &p,$FR$\"38hg8XVi\"R\"F_x7$$\"3q+D\"yD%fOIFR$\"3XAVq4B*p)HF^w7$$\"3jvV BSYBcIFR$\"36w7jB#))[E\"F^w7$$\"3,^ilA](e2$FR$\"3nZ7T?-/[pF)7$$\"3&f7y ]S:b4$FR$\"3f?q#)p+V%Q%F)7$F]gl$\"3CJK#fm'R;IF)7$$\"3N8GQn)eT8$FR$\"3v B&zb'[RAAF)7$$\"3#eils%>;`JFR$\"3wz'Q`*G80$FR$\"3/]ZoMO`%4\"F)7$$\"3AjS\"p=r,@$FR$\" 3MO)>w2Db0*F17$$\"3pvozmUVLFR$\"3Uu g4xf7'R$F17$$\"3P+]il]?>MFR$\"3YM'4[!==wAF17$$\"37+vo/(H7d$FR$\"3eL\"f k3`eA\"F17$Fbgl$\"3QEI]xvwYwFR7$$\"3X*\\Pfy^?*QFR$\"3K-%['4SmE]FR7$$\" 3/**\\7G#\\31%FR$\"38!R^7Q)GaNFR7$$\"3k)\\7.nY'HUFR$\"3^tmhA&[ak#FR7$F ggl$\"3\"Q%or9+OX?FR7$$\"31)**\\78)y,ZFR$\"3GYfOff<\"Q\"FR7$F\\hl$\"3D 84^o`()[**F-7$Fahl$\"3Ep\"[H;ENo&F-7$Ffhl$\"3gJC&H@'o5QF-7$F[il$\"3F-7$Feil$\"3)\\Ib#)eMz_\"F-7$Fjil$\"3^(pr zh^n@\"F-7$F_jl$\"3&30>XGw!G)*Fcdl7$Fdjl$\"3@rS:'HA!f\")Fcdl7$Fijl$\"3 tTG3,1IcoFcdl7$F^[m$\"3Riy+D2$)\\eFcdl7$Fd[m$\"38OLJZEK3^Fcdl7$Fi[m$\" 3WFcdl7$F^\\m$\"3X:jG(o&3:RFcdl7$Fc\\m$\"3ZDd\\$RZnY$Fcdl7$Fh \\m$\"3%QGS;'H01JFcdl7$F]]mFfin-Fb]m6&Fd]mFe]mFh]mFe]m-F$6$7eq7$F(F^in 7$F/$\"3WzrP')[pi'*Fb[m7$F5$\"3Z63D7kmA5Fcdl7$F:$\"3)pl)f-k>#4\"Fcdl7$ F?$\"3*z\\A0R-'p6Fcdl7$FD$\"3yX$\\;-3^D\"Fcdl7$FI$\"3'R#y'fJWIM\"Fcdl7 $FT$\"3&\\7\"))yb?W9Fcdl7$Fhn$\"3J@p>K.Mh:Fcdl7$Fjq$\"33/ZJyh(Gp\"Fcdl 7$F]\\l$\"3^Nl;&>4j%=Fcdl7$Fc`l$\"3e(Q55*QT**>Fcdl7$F]al$\"3V7qtyYr&># Fcdl7$Fgal$\"3lTz&o77MU#Fcdl7$Fabl$\"32zGMt_3yEFcdl7$Ffbl$\"3(Q&zl*43e %HFcdl7$F[cl$\"30/7*e>s)>LFcdl7$F`cl$\"3*3*HR8bX&p$Fcdl7$Fecl$\"3L.%eh jF%=UFcdl7$Fjcl$\"3Ftzt/([)zZFcdl7$F_dl$\"3%4U%>jT:NbFcdl7$Fedl$\"3Yf7 1BW*RV'Fcdl7$Fjdl$\"3g(Rpzhvpi(Fcdl7$F_el$\"3'Hwf>q%)3/*Fcdl7$Fdel$\"3 Ce#=F17$ $\"3)*)*\\7.+$>d&FR$\"3AD55nF)G=#F17$$\"3J*\\iSc:Ch&FR$\"39$f!**Q7tiEF 17$Fahl$\"3@=(y(e%=,K$F17$$\"3mOMx#4W7n&FR$\"3>o6ffp%4q$F17$$\"3euoagq e*o&FR$\"3rj'340k7:%F17$$\"3\\7.KG+$zq&FR$\"3F]8h\\T1*o%F17$$\"3^\\P4' *HFEdFR$\"3$)*y=Y2n&Q`F17$$\"3a'=nQ'fhWdFR$\"3EcDX=M*H8'F17$$\"3XC1kJ* eHw&FR$\"3my)=(y%)*)=rF17$$\"3OiST**=I\"y&FR$\"3'z-7Gf^JO)F17$$\"3R*\\ (=n[k*z&FR$\"3a(H'y%3\"ek**F17$$\"3TO4'\\$y)z\"eFR$\"3Yj,Y4AU27F)7$$\" 3KuVt-3LOeFR$\"3)G)>;oyA$\\\"F)7$$\"3C7y]qPnaeFR$\"3Obe*H8lR*=F)7$$\"3 E\\7GQn,teFR$\"3^#eGh!)e1[#F)7$$\"3G'oagqf8*eFR$\"3>#HZ$z60*Q$F)7$$\"3 ?C\"GQn-(4fFR$\"3`)yeq>/e!\\F)7$$\"36i:gTc/GfFR$\"34,Ksr7%es(F)7$$\"38 **\\P4')QYfFR$\"3*4Q))=q+J&p7vT,gFR$\"3..AZ%>,2*>! #67$$\"3,\\(o/[g(>gFR$\"3e:34+#*QC5F_x7$$\"3.'=U#[M5QgFR$\"3112.]B1bFF ^w7$$\"3&Ri:gTYk0'FR$\"3..Yn2[Tb7F^w7$$\"3'=1*y$Q*yugFR$\"3z*GeO0>7:(F )7$$\"3)))\\i:NKJ4'FR$\"3f5Z8*pw;h%F)7$$\"3\"f$fL>`Z6hFR$\"3/_f)yNj)=K F)7$$\"3#QP4rG=)HhFR$\"3p2&\\)oX\\tBF)7$$\"3t6G)[Dh\"[hFR$\"3+Z[='FR$\"3E6CkX%p1< \"F)7$$\"3pBJ?e,>.iFR$\"3cHmhayX)o*F17$$\"3hhl(f7L:A'FR$\"3n10x\\(p/:) F17$Ffhl$\"37VYy,0i^pF17$$\"3^h!RD!)H*fiFR$\"3WC:Z3iN?fF17$$\"3QC\"G8^ $)*ziFR$\"3/a-`ogj FR$\"3CU=%*e:-$3$F17$$\"3(pVt_0_-Q'FR$\"3OBRsW>TmFF17$$\"3&)*\\iSw0.S' FR$\"3gW!yrK#='\\#F17$$\"3gD1k\"=8/W'FR$\"3I)R.HzSA1#F17$$\"3X](=#*f?0 ['FR$\"3$*pj@?8NK1hJDzd&GFR7$F`il$\"3O3Ei-@*G\"=FR7$$ \"3^++DJ$y4!yFR$\"3Q%*=QwpAL7FR7$Feil$\"3:HTieJ'*G*)F-7$$\"3c**\\(=7)3 D%)FR$\"3<7#)QfP],oF-7$Fjil$\"3hh=Mmn#GN&F-7$F_jl$\"3Spo\\'>P?]$F-7$Fd jl$\"350cdwms(\\#F-7$Fijl$\"3$=4+*[NVf=F-7$F^[m$\"3a$*[Z)*QuS9F-7$Fd[m $\"3F,*>#[([#p6F-7$Fi[m$\"35EY\"yQ/$G%*Fcdl7$F^\\m$\"3>=9h5nQumFcdl7$Fh\\m$\"3K!G'y%)RbRdFcdl7$F]]mF]^m-Fb]m6&Fd]mFh ]mFe]mFe]m-F$6$7eq7$F(F_]m7$F/$\"3k_,j8\\\"*QtFb[m7$F5$\"3BneZHM95xFb[ m7$F:$\"3(zA7Cz)Hi\")Fb[m7$F?$\"3by7OLM%)e')Fb[m7$FD$\"3k%GP*pST*>*Fb[ m7$FI$\"34fM)z[Bru*Fb[m7$FT$\"3m$H<%pVuO5Fcdl7$Fhn$\"3^'=U%f1M26Fcdl7$ Fjq$\"3WJW3bx8&=\"Fcdl7$F]\\l$\"3;n\\4(>QSF\"Fcdl7$Fc`l$\"3]1%4(zl*3O \"Fcdl7$F]al$\"3'oGAIpX(p9Fcdl7$Fgal$\"3rcvMg*HFf\"Fcdl7$Fabl$\"3%3!\\ 4GuXEPDiH_7mFcdl7$Fcfl$\"3D#4T!Q&Q<\"yFcdl7$Fhfl$\"3*eN(eQ\\sG%* Fcdl7$F]gl$\"3U'*)3CX@]:\"F-7$Fbgl$\"3@*=1u*GdO9F-7$Fggl$\"3DaQ8S53*)= F-7$F\\hl$\"3\\&HW]bFk]#F-7$Fahl$\"3e#fjag\\/d$F-7$Ffhl$\"3B*ob#\\i_]_ F-7$Fi\\t$\"3.R7QXfnAnF-7$F[il$\"3z]Swi&pL\"*)F-7$Ff]t$\"3>M)[n[RA@\"F R7$F`il$\"3>__#e'QnVXnj$F)7$$\"3%p$4@\" \\7`\"*)FR$\"3v-;7Z\\ExbF)7$$\"3jtVBl.]N*)FR$\"3$*)**=?p:\\h*F)7$$\"3K 5yDR#)ob*)FR$\"3P:K@i^9P?F^w7$$\"3z[7G8h(e(*)FR$\"3Vt%3R1-L(oF^w7$$\"3 E(o/t)R1'**)FR$\"3;WwSR^%>e#Fiw7$$\"3&R7G8'=D;!*FR$\"3]T0OSfW9:F_x7$$ \"3kg:NN(Rk.*FR$\"3q&\\7TmyB,$F^w7$$\"36**\\P4wic!*FR$\"3%yL+4n$RZ7F^w 7$$\"3eP%)R$[:o2*FR$\"3G`l#zQ]*ynF)7$$\"3Fu=UdL+(4*FR$\"39&y%4EX&4D%F) 7$$\"3'4JX9B\"><\"*FR$\"3(*Rx\")Gc_7HF)7$$\"3V\\(oa5zt8*FR$\"3,DhT#yF% >@F)7$$\"3!z=#\\zpcd\"*FR$\"3I*G.GCE6h\"F)7$$\"3fCc^`[vx\"*FR$\"3+lmBl 9&fE\"F)7$$\"3Gh!RvsUz>*FR$\"3AFUgiX*3-\"F)7$$\"3v*\\i:gI\"=#*FR$\"3)Q 776:CnS)F17$$\"3AQfev%=$Q#*FR$\"36a7&*4syUqF17$$\"3\"\\P4'\\j]e#*FR$\" 316u%*Rgt&)fF17$$\"3g6GjBUpy#*FR$\"31DT&)zB'*\\^F17$$\"32]il(4#)))H*FR $\"3cU6gUNvxWF17$$\"3a)oz;(*p!>$*FR$\"3!)4z\\S^0HRF17$$\"3BDJqXyDR$*FR $\"3S\"Q*)e9j`Z$F17$$\"3#>cE(>dWf$*FR$\"3\"pYCz'e%f4$F17$F_jl$\"3<2.X; GVvFF17$$\"3UD1*yB?&=%*FR$\"377$[_'pj$G#F17$$\"3Y]7.#)oSd%*FR$\"3*H3RL Y`=\">F17$$\"3]v=&e&FR7$Fdjl$\"3rO)4#[/[&)RFR7$$\" 3:+]7ed*>.\"F1$\"3Ob%4lmJeH#FR7$Fijl$\"3=&od$)eq1\\\"FR7$$\"3>+DJg)f`4 \"F1$\"3a-AUdw1[5FR7$F^[m$\"3;L?QT[woxF-7$Fd[m$\"3`T[-nmGG\\F-7$Fi[m$ \"3(RCDBM&HSKF-7$F^\\m$\"3Jj9Wh[VqBF-7$Fc\\m$\"3JO8aF;9z " 0 "" {MPLTEXT 1 0 37 "equations(essential_components(p,R));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7#,&*$)-%%diffG6$-%\"yG6#%\"tGF.\"\"#\"\" \"F0*$)F+\"\"$F0!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 38 "It appears that the singular solution " } {XPPEDIT 18 0 "y(t) = 0;" "6#/-%\"yG6#%\"tG\"\"!" }{TEXT -1 266 " does not belong to this minimal description. And indeed, the graph of the \+ non-singular solutions along the zero function is very much alike in t he previous example: the non-singular solutions collapse on the zero f unction. In both cases, the singular solution is a " }{TEXT 271 5 "lim it" }{TEXT -1 119 " of the non-singular solution. It is in that sense \+ that the singular solution here is embedded in the general solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 8 " \+ " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 38 "Clairaut partial differenti al equation" }}{PARA 0 "" 0 "" {TEXT -1 114 "Similar enveloping and li mitting properties can be found for singular solutions of partial diff erential equations." }}{PARA 0 "" 0 "" {TEXT -1 38 "Consider first a p de of Clairaut type:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "pde \+ := -y(t,s)+t*diff(y(t,s),t)+s*diff(y(t,s),s)-diff(y(t,s),t)^2 -diff(y( t,s),s)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG,,-%\"yG6$%\"tG% \"sG!\"\"*&F)\"\"\"-%%diffG6$F&F)F-F-*&F*F--F/6$F&F*F-F-*$)F.\"\"#F-F+ *$)F2F6F-F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Q := differe ntial_ring(ranking=[y],derivations=[s,t],notation=diff):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "equations(Rosenfeld_Groebner( [pde] , Q));" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%7#,,-%\"yG6$%\"sG%\"tG\"\"\"*&F*F+-%%diffG6$F&F*F+!\"\"*&F)F+- F.6$F&F)F+F0*$)F-\"\"#F+F+*$)F2F6F+F+7$,&F)F0*&F6F+F2F+F+,*F&\"\"%*(F= F+F*F+F-F+F0*&F=F+F5F+F+*$)F)F6F+F07#,(F&F=*$)F*F6F+F0F@F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "The solutions are:" }}{PARA 0 "" 0 "" {TEXT -1 15 " " }{XPPEDIT 18 0 "y(t,s) = a*t+b*s-a^2-b^2 " "6#/-%\"yG6$%\"tG%\"sG,**&%\"aG\"\"\"F'F,F,*&%\"bGF,F(F,F,*$F+\"\"#! \"\"*$F.F0F1" }{TEXT -1 18 " " }{XPPEDIT 18 0 "y(t,s) = a*t+1/4*s^2-a^2 " "6#/-%\"yG6$%\"tG%\"sG,(*&%\"aG\"\"\"F'F,F,*(F, F,\"\"%!\"\"F(\"\"#F,*$F+F0F/" }{TEXT -1 13 " " }{XPPEDIT 18 0 "y(t,s) = (t^2+s^2)/4" "6#/-%\"yG6$%\"tG%\"sG*&,&*$F'\"\"#\"\"\"* $F(F,F-F-\"\"%!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 129 "See in the animation below the enveloping property of \+ the singular solution. The moving plane is a non-singular surface inte gral." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "equations(essential_components(pde,Q));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7$7#,,-%\"yG6$%\"sG%\"tG\"\"\"*&F*F+-%%diffG6$F& F*F+!\"\"*&F)F+-F.6$F&F)F+F0*$)F-\"\"#F+F+*$)F2F6F+F+7#,(F&\"\"%*$)F*F 6F+F0*$)F)F6F+F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "plots[animate3d](\{(t^2+s^2)/4, a/ 2*t-(a/4)^2*s-(a/2)^2-(a/4)^4\},t=-4..4,s=-4..4, a=-5..3,orientation=[ 45,111]):" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 28 "Particular solutio n of a pde" }}{PARA 0 "" 0 "" {TEXT -1 107 "We now illustrate the poss ible limiting property of a singular solution to a partial differentia l equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 51 "pde := diff(y(s,t),s)^2-diff(y(s,t),t)^2+ y(t,s)^3; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pdeG,(*$)-%%diffG6$-%\"yG6$%\"s G%\"tGF.\"\"#\"\"\"F1*$)-F)6$F+F/F0F1!\"\"*$)-F,6$F/F.\"\"$F1F1" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 39 "The general solution can be given as: \+ " }{XPPEDIT 18 0 "y(s,t )= 4*(-1+b^2)/((b*t+s -c)^2)" "6#/-%\"yG6$%\"s G%\"tG*(\"\"%\"\"\",&F+!\"\"*$%\"bG\"\"#F+F+*$,(*&F/F+F(F+F+F'F+%\"cGF -F0F-" }{TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "There is furt hermore a singular solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Q := differential_ring(ranki ng=[y],derivations=[s,t],notation=diff):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "equations(Rosenfeld_Groebner([pde],Q));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7%7#,(*$)-%%diffG6$-%\"yG6$%\"sG%\"tGF.\"\"#\"\" \"F1*$)-F)6$F+F/F0F1!\"\"*$)F+\"\"$F1F17$F(,&F2F1F7F67#F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "Nonethele ss, the singular solution does not belong to the minimal description o f the solution set." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "equations(essential_components(pde,Q));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7#,(*$)-%%diffG6$-%\"yG6$%\"sG%\"t GF.\"\"#\"\"\"F1*$)-F)6$F+F/F0F1!\"\"*$)F+\"\"$F1F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 175 "Indeed, the si ngular solution is embedded in the general solution, as shown in the f ollowing animation where non-singular solutions collapse smoothly on t he singular solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "plots[animate3d](\{0, -4*(-1+(1+1/ (c/2))^2)/((1+1/(c/2))*t+s-c/2)\},s=-1..0,t=-1..0,c=1..10,orientation= [45,69]):" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 35 "Higher order diffe rential equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 " " 0 "" {XPPEDIT 18 0 "(6*y*diff(y,t,t)-5*diff(y,t)^2)^3+729*y^4" "6#,& *$,&*(\"\"'\"\"\"%\"yGF(-%%diffG6%F)%\"tGF-F(F(*&\"\"&F(*$-F+6$F)F-\" \"#F(!\"\"\"\"$F(*&\"$H(F(*$F)\"\"%F(F(" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "ode := (6*y(t)*diff(y(t),t,t)-5*diff(y(t),t)^ 2)^3+729*y(t)^4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG,&*$),&*&-% \"yG6#%\"tG\"\"\"-%%diffG6$F*-%\"$G6$F-\"\"#F.\"\"'*&\"\"&F.)-F06$F*F- F5F.!\"\"\"\"$F.F.*&\"$H(F.)F*\"\"%F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "equations(Rosenfeld_Groebner([ode],R));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7$7#,,*&)-%\"yG6#%\"tG\"\"$\"\"\")-%%diffG6$F(-% \"$G6$F+\"\"#F,F-\"$;#**\"$S&F-)F(F5F-)F/F5F-)-F06$F(F+F5F-!\"\"**\"$] %F-F(F-F/F-)F<\"\"%F-F-*&\"$D\"F-)F<\"\"'F-F>*&\"$H(F-)F(FBF-F-7#F(" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "plot( \{seq(seq(((x-a)+b/2 *(x-a)^2)^3, a=-1..1), b=2..4)\}, x=-2..12/10, -2/100..2/100);" }} {PARA 13 "" 1 "" {GLPLOT2D 355 318 318 {PLOTDATA 2 "6--%'CURVESG6$7Z7$ $!\"#\"\"!$\"$;#F*7$$!3immmwAc#)>!#<$\"3YfPJs\\YJ?!#:7$$!3DLLL`X7l>F0$ \"3n=n9XEM4>F37$$!3()******HooZ>F0$\"3%z)z@LbP$z\"F37$$!3rmmm1\"\\-$>F 0$\"3a,z\\^RJ$o\"F37$$!3()*****p;/***=F0$\"31YfEB2:0:F37$$!3CLLLF#f&p= F0$\"3F>ui^*>HM\"F37$$!3')******oJVN=F0$\"3tSyZoH2y6F37$$!3rmmm5rI,=F0 $\"3a[hRn(o..\"F37$$!3!ommE%[&pw\"F0$\"3$eS-M'fJv*)!#;7$$!3nmmmuDgKM(G#Ffn7$$!3!******f +J'*R\"F0$\"3Pfhe(zpqf\"Ffn7$$!3DLLLlZ-I8F0$\"3BMRteAKw5Ffn7$$!3vmmm]b ro7F0$\"3rf>l\"Q'H@uF07$$!3$******f*ep*>\"F0$\"3V[O7Pc&=t%F07$$!32+++S GRI6F0$\"3*ptGQAcZ*GF07$$!3,+++/lgj5F0$\"3\"Q<)*=;+Ls\"F07$$!3ommm7\"e H+\"F0$\"38L(4ST8p-\"F07$$!3AMLL$48%3$*!#=$\"3u%R!\\@'G-;&F[s7$$!3gKLL t\"*[(p)F[s$\"32zsTi&y1m#F[s7$$!3O+++?%Ro)zF[s$\"31'HtCi^g3\"F[s7$$!3H NLLtRzdtF[s$\"3NX*oxxko<%!#>7$$!3v******>9jnmF[s$\"32jQ=(zx(*4\"F]t7$$ !3y******fMV5gF[s$\"3Xf]2O\"p>z\"!#?7$$!3]KLLLjrC`F[s$\"3/'p,fCb^8%!#A 7$$!3\\LLL`R,&p%F[s$!3$zKzXPq([BF^u7$$!3_mmmm[z:SF[s$!3!z2%zRZDR\\!#@7 $$!3^nmmErF5LF[s$!35m'4iq/+S\"Fht7$$!3J+++g=7'p#F[s$!3E')RommG<>Fht7$$ !3emmmYt\"G.#F[s$!34ge22!pbv\"Fht7$$!3d,++!))evM\"F[s$!3&)[_wI?cQ&*Fiu 7$$!3c1+++;nrnF]t$!3F$>(zPCq1?Fiu7$$!3P=,+++(G&GFht$!3=\\G2jYR#G#!#D7$ $\"3P*)*****>Pn\"pF]t$\"3g(y+^5c5)[Fiu7$$\"3kimmEw!)Q8F[s$\"3=\\dua8`* )[Fht7$$\"3/-+++7wH?F[s$\"3#z\")\\)ed0CBF]t7$$\"3(=LLL$y'el#F[s$\"3OyI XOb+DnF]t7$$\"3N+++gxOSLF[s$\"3\"fw*QR'H*HmmmmI'eJ&F[s $\"38S8F07$$\"3u'******\\U\\+'F[s$\"31+\"=gzl(3BF07$$\"3#fLLL*yg omF[s$\"3^*3iqGI#pPF07$$\"3VKLL8;IZtF[s$\"3%ooph:iH(fF07$$\"3#)omm1aP? !)F[s$\"3zQ_A$)*\\06*F07$$\"3:******>.')Q')F[s$\"3)[\"e_T)f&38Ffn7$$\" 3Ffn7$$\"3EJLL`3s\")**F[s$\"3%3a;u=;an#Ff n7$$\"3D+++e/xl5F0$\"3t(*4\"pkAwr$Ffn7$$\"3)******fsq/8\"F0$\"3()4$f%o ^k4]Ffn7$$\"3'**************>\"F0$\"3(4+++?Jc!*)GFd^l7$FD$\"3!p9<2%yj'p#Fd^l7$FI$\"3N)\\U6j,]^#Fd^l7$FN$\"3bbxP ]s+BBFd^l7$FS$\"3$H@;E#QOV@Fd^l7$FX$\"3gC!48-`V(>Fd^l7$Fhn$\"3+U'f0Twl \"=Fd^l7$Fbo$\"3=,2Gd1fL:Fd^l7$Fgo$\"3*4A,$>B'\\I\"Fd^l7$F\\p$\"3#[c!4 tt$))4\"Fd^l7$Fap$\"3n_S\"zo#))[\"*F37$Ffp$\"3pVwSt/fxvF37$F[q$\"3]UC# oL`@?'F37$F`q$\"3DC[<47(*o^F37$Feq$\"3AtQVV+6\"=%F37$Fjq$\"31u'3H#HK_L F37$F_r$\"3A<>pf$Gvo#F37$Fdr$\"3mu<,J'fC=#F37$Fir$\"3\"*3^Y.v&po\"F37$ F_s$\"31`m-4q)QM\"F37$Fds$\"3w'y6A^#o>5F37$Fis$\"3y52[<_!e*yFfn7$F_t$ \"3\"*)yi$4^#Q)eFfn7$Fdt$\"3TVN;l\\S#Q%Ffn7$Fjt$\"3y#H(pp,&)oJFfn7$F`u $\"3-g&*>E6P8BFfn7$Feu$\"3&yNs8A#>9;Ffn7$F[v$\"35[*G9BuE3\"Ffn7$F`v$\" 3g<]#f@'4juF07$Fev$\"3jJCUKQ>[[F07$Fjv$\"3Z`bDS1j*)HF07$F_w$\"3YT%47Aw #\\`1S*[F^u7$F]z$!3We;.=p/\"f#F^u7$ Fbz$!3GSJ]?W0x^Fiu7$Fgz$!3ief%**GPTP\"Fht7$F\\[l$!30K<3_pMJ>Fht7$Fa[l$ !3Uj^dC\"*457%Fiu7$Fj\\l$\"3xnOU9eq_WFht7$F_]l $\"3&y*********>&>#F]t-Fd]l6&Ff]lFj]lFg]lFj]l-F$6$7X7$F($\"3+++++]id6F d^l7$F5$\"3VUPk)zY$o5Fd^l7$F?$\"3AkB-I?k[)*F37$FD$\"3SNUM\\o/n\"*F37$F I$\"3$HT3N_'4D&)F37$FN$\"3Q&yfyN1#[yF37$FS$\"3$=N2Lqel@(F37$FX$\"3rqY) H-WRi'F37$Fhn$\"3xE-LErIsgF37$F]o$\"3d4Q#HX2F37$F`q$\"3*=w'**33Z?;F37$Feq$\"3k+tEK YB%H\"F37$Fjq$\"3$))y]nFpL-\"F37$F_r$\"3\"4_N;3S^3)Ffn7$Fdr$\"3a%\\q0, @8Z'Ffn7$Fir$\"3A%G&fB7M3\\Ffn7$F_s$\"3fGn:svdTQFfn7$Fds$\"39)\\+%>K. \\GFfn7$Fis$\"3zX!\\&f$*)o:#Ffn7$F_t$\"31B5)GqBKc\"Ffn7$Fdt$\"3wXf)=WB *H6Ffn7$Fjt$\"3VKfa&F07$Feu$\"3vEm_&pp$)o $F07$F[v$\"3*R#)fzc8PL#F07$F`v$\"3T;[:UQ)R^\"F07$Fev$\"3ZIXTEW#e3*F[s7 $Fjv$\"3Tv$y$zE'y0&F[s7$F_w$\"39l%3*p'\\*\\EF[s7$Fdw$\"3lQts0pP$H\"F[s 7$Fjw$\"3XGz_Sp'y,&F]t7$F_x$\"3&)eFgSe!*R6vxcTFiu7$F^y$!3\"Ro\"Fht7$F]z$!3Jn>$o=[Gq#Fht7$Fbz$!343M2&>oN5%Fht7$Fgz$!3?7+ =>\\iHYFht7$F\\[l$!3.Sh6!=eV2%Fht7$Fa[l$!3s`,D'e$*fp#Fht7$Ff[l$!3gyW0y S1r7Fht7$F[\\l$!3]liNetzP?Fiu7$F`\\l$!3PV#eMKfu0'Fegl7$Fe\\l$\"3_t%))* )[=Hx$Fiu7$Fj\\l$\"30S&px([v'z$Fht7$F_]l$\"30)*********fdNaGF^u7 $Ffp$!37r^'GGtD;&Fiu7$F[q$!3NVPnE&o@T\"Fht7$F`q$!3@)oVqqq/#>Fht7$Feq$! 3f*H>3*eODbs+S$R^&*)Fiu7$F_r$!3K/6A+(=4r\"Fiu7$Fdr$!3oek% 4>0Ga#Fhw7$Fir$\"3)=H6E=&**y[Fiu7$F_s$\"3.iT=gqiDWFht7$Fds$\"3v-eX.AZ^ AF]t7$Fis$\"3zM#>S)=P'e'F]t7$F_t$\"3l1zl8Vf7Ffn7$Fdw$\"3)**QVCi7F37$F]z$\"3w!*=)ev<+T#F37$Fbz$\"3!p^*exFgRIF37$Fgz$\"3Ksq=ZiZpPF37$ F\\[l$\"3C'y0PA13m%F37$Fa[l$\"3]X\\LCf36dF37$Ff[l$\"3X:Rw)Q-F%oF37$F[ \\l$\"3lS;B/[$GO)F37$$\"3r******Rtrk'*F[s$\"3$>(\\3*=C\"G\"*F37$F`\\l$ \"3\\LV7Muu]**F37$$\"3pmmmr7(>.\"F0$\"3S`:eR$z%*3\"Fd^l7$Fe\\l$\"3T&p! \\9M@\">\"Fd^l7$$\"37+++#f?\")4\"F0$\"3\"ow3'\\M'eH\"Fd^l7$Fj\\l$\"3B9 ,.&*o/39Fd^l7$$\"3')*****HON_;\"F0$\"3[W!pu,*[P:Fd^l7$F_]l$\"3>,++smnw ;Fd^l-Fd]l6&Ff]lFj]lFj]lFg]l-F$6$7W7$F($\"3+++++++]7F[s7$F?$\"3j.T&=g> `(\\F]t7$FI$\"3YzD@,@S`=F]t7$FS$\"3T2\"=XTb$QUFht7$Fhn$\"3*[BY*z*RS!QF iu7$Fbo$!3(Q`AEBj/W\"Fhw7$Fgo$!3(HD6kI=!*3#Fiu7$F\\p$!3S!>@([.)e<\"Fht 7$Fap$!3j[Ac%Q(3HFFht7$Ffp$!3O7$eFma;5%Fht7$F[q$!3dyflO9EGYFht7$F`q$!3 [,$*)\\Ylq7%Fht7$Feq$!3^s?BWb%ot#Fht7$Fjq$!3?fP/(yqR:\"Fht7$F_r$!3E0eX %yZ[!>Fiu7$Fdr$!31_X3#4sTb#Fhw7$Fir$\"3$*e[Qn*GxW%Fiu7$F_s$\"3TY69+-\\ uPFht7$Fds$\"3&**QJix.2!=F]t7$Fis$\"3mIMu7c'=-&F]t7$F_t$\"3cEg+LNb[7F[ s7$Fdt$\"3P-zFgbN$f#F[s7$Fjt$\"3mR@=J[BK]F[s7$F`u$\"35`+e2ZZX')F[s7$Fe u$\"3[Q!)=&f#Qk9F07$F[v$\"31zWgt\\\\2CF07$F`v$\"3JQiQ#Q.de$F07$Fev$\"3 nIwqw?*)[`F07$Fjv$\"3b*zJ1s8%fyF07$F_w$\"3hi0IC#\\z6\"Ffn7$Fdw$\"3x\\X )\\nB7a\"Ffn7$Fjw$\"39)ovm\"QUd@Ffn7$F_x$\"3am%RCRM?(GFfn7$Fdx$\"3qb%) oVq!*RQFfn7$Fix$\"3yaoPe9iN\\Ffn7$F^y$\"3&[+bB/,pT'Ffn7$Fcy$\"3shQ%yi- !H\")Ffn7$Fhy$\"3mdbf0ygI5F37$F]z$\"3*fDaS^X!)G\"F37$Fbz$\"3ah@#=aQCh \"F37$Fgz$\"3tac=+I9')>F37$F\\[l$\"3`3.<$4S)RCF37$Fa[l$\"3=:n#R`x9(HF3 7$Ff[l$\"3k*o!HqXUTNF37$F[\\l$\"3_Y(\\ko:LI%F37$Fd[n$\"3lu%[:z@bo%F37$ F`\\l$\"3$*)4-9p\"[&4&F37$F\\\\n$\"3eXf%z-Q\\c&F37$Fe\\l$\"3f;0%HqX(pg F37$Fd\\n$\"3t+nU)o\\ze'F37$Fj\\l$\"3B=Zs&[iB9(F37$F\\]n$\"3:9'fKk\"z! y(F37$F_]l$\"3q-++g`!fY)F3-Fd]l6&Ff]lFg]lFj]lFg]l-F$6$7gn7$F($\"#kF*7$ F.$\"3hO\\Uidk#*fFfn7$F5$\"3Q$yqx?ipg&Ffn7$F:$\"3FOGrEY-U_Ffn7$F?$\"3: 8%QZ(z$p*[Ffn7$FD$\"3[j%)\\eB>TVFfn7$FI$\"3x25t;AfQQFfn7$FN$\"3!3?$)p! GvJLFfn7$FS$\"3A'G4`.b9)GFfn7$FX$\"3\"zQ]lc,,[#Ffn7$Fhn$\"3qCO1MK+E@Ff n7$F]o$\"3,d6(3rKg\"=Ffn7$Fbo$\"3y$>w'fCFW:Ffn7$Fgo$\"3cq(3')*\\pG6Ffn 7$F\\p$\"3PXW\"H;e,+)F07$Fap$\"3D%G6'px#=Z&F07$Ffp$\"38F[Qr3(Qk$F07$F[ q$\"3[b&fzl7zJ#F07$F`q$\"3*o8RTZ[S]\"F07$Feq$\"3nc8Ut8_D))F[s7$Fjq$\"3 8YarHqBh[F[s7$F_r$\"3F\"\\()y'*[(RDF[s7$Fdr$\"3sWJ)\\d**\\H\"F[s7$Fir$ \"3gb#zs2-&=]F]t7$F_s$\"3#pKO*e>#)f=F]t7$Fds$\"36A!)eCcJcRFht7$Fis$\"3 C$p%\\?e.QWFiu7$F_t$\"3Gw+3K3L$)*)!#I7$Fdt$!3!y(ow_;$42#Fiu7$Fjt$!3_]b ^OXKJ7Fht7$F`u$!3OD+8p%orn#Fht7$Feu$!3fl*=]p+:2%Fht7$F[v$!3F&y(ev^'*GY Fht7$F`v$!3)*zdO&=./9%Fht7$Fev$!3*=$*z)4Z%4#GFht7$Fjv$!3g?x`B?*GC\"Fht 7$F_w$!3,9kQfWohqR`Zp[\"F07$Fgz$\"3t)zw:*zZtBF07$F\\[l$\"3&Hm5Z UzTo$F07$Fa[l$\"3pvmG9_Y;bF07$Ff[l$\"34%[ZE`x;!yF07$F[\\l$\"3(eWw(4H>K 6Ffn7$F`\\l$\"3thYjQ+%)[:Ffn7$F\\\\n$\"3><=$4'z$z\"=Ffn7$Fe\\l$\"3;u'H ?r/W7#Ffn7$Fd\\n$\"3E(f>()f5kX#Ffn7$Fj\\l$\"3(H%y'Hff+$GFfn7$F\\]n$\"3 M(=H#[li#G$Ffn7$F_]l$\"3x++++cI$z$Ffn-Fd]l6&Ff]lFj]lFg]lFg]l-F$6$7fn7$ F($\"\")F*7$F5$\"37lj7ua8AoF07$F?$\"3LS3ENuX*y&F07$FD$\"3EU!zvbdy*\\F0 7$FI$\"3qmki!o.lH%F07$FS$\"3h&>V_#f=2IF07$Fhn$\"3-qBr%zT]/#F07$Fbo$\"3 CVB/jvw]8F07$Fgo$\"3(H+I0IDz*))F[s7$F\\p$\"3\\tV1>6GYbF[s7$Fap$\"3%y5x #4:jBKF[s7$Ffp$\"3PXaF%**>*\\ZqQX\"Fht7 $Fds$!3:(*e3.%4o:%Fht7$Fis$!3-/Bl2gbZtFht7$F_t$!3P@$z'*>BOwFht7$Fev$!3xJ*e*RMC[ UFht7$Fjv$!3o*\\a+G4^e\"Fht7$F_w$!3q[>?T+7;DFiu7$Fdw$!3&GOc(3t4-BFhw7$ Fjw$\"39X>)f?xU/%Fiu7$F_x$\"3J(RWi8(H)\\$Fht7$Fdx$\"3G6KA*)Q\"eX\"F]t7 $Fix$\"3B6EA]rZ(z$F]t7$F^y$\"3(3IB#\\#Q)[))F]t7$Fcy$\"3A?qBDb%*H=$F[s7$F]z$\"3[neO7v\"pR&F[s7$Fbz$\"3jr*3g)QYx))F[s7$F gz$\"3*[WQ%R0Ut8F07$F\\[l$\"3j)))*oF)>02#F07$Fa[l$\"3LEQ:]n3>IF07$Ff[l $\"35951B1suTF07$$\"3mLLLtqG$**)F[s$\"3-)HbgZWP)\\F07$F[\\l$\"3?_c=Rqo :fF07$Fd[n$\"3z&>i@yU['oF07$F`\\l$\"3'R#ea6_TMzF07$F\\\\n$\"3CSLj&4A0A *F07$Fe\\l$\"3X#Qwz()yr1\"Ffn7$Fd\\n$\"3'Q$[.3A/B7Ffn7$Fj\\l$\"37iw8*f EqR\"Ffn7$$\"3#*****\\WI&y9\"F0$\"39>P^qXb)\\\"Ffn7$F\\]n$\"3M(GX))zRg g\"Ffn7$$\"3-++]\"oF37$F?$\"3x9gxl'z%4=F37$FD$\"3=KpT&eACn \"F37$FI$\"3=b@Enn0W:F37$FN$\"3isb/k)G&49F37$FS$\"3nlLZ'QH[G\"F37$FX$ \"3<])yS*[ko6F37$Fhn$\"3X2l%G(4Fh5F37$F]o$\"3>JbBlDAE'*Ffn7$Fbo$\"3G

wHFfn7$F`q$\"3XQVX. bp%Q#Ffn7$Feq$\"3,FM*oc8y$=Ffn7$Fjq$\"3An\"Qco%e'R\"Ffn7$F_r$\"3<#e>@! =Od5Ffn7$Fdr$\"378\\o<722\")F07$Fir$\"3/e+:`<'e!eF07$F_s$\"3(=Woi'[i+V F07$Fds$\"3[WP5O#fZ'HF07$Fis$\"3*y(G%\\OyJ3#F07$F_t$\"3^!oUL0wDP\"F07$ Fdt$\"3Kd7SYU+6*)F[s7$Fjt$\"30H'p%*[QLV&F[s7$F`u$\"3Y:,@(\\KTG$F[s7$Fe u$\"3I!e\"[%RnIy\"F[s7$F[v$\"31%)oimmq`&)F]t7$F`v$\"3$ok%yEt!3,%F]t7$F ev$\"3,\\C%)eX^j9F]t7$Fjv$\"3?iKxSOhvNFht7$F_w$\"3'*zV%4g'ozPFiu7$Fdw$ \"39D<_7@%=M#Fhw7$Fjw$!32*=$3:B=r6D+ %Fht7$Ff[l$!3Kjt/Q'Qei\"Fht7$F[\\l$!3flAJ[:*oE#Fiu7$F`\\l$!3h[/)Q.HT2' Fegl7$Fe\\l$\"3E%GEh[iTW$Fiu7$Fj\\l$\"3^[O3m>h3KFht7$F_]l$\"3\"*)***** ****R#Q\"F]tF_hl-F$6$7W7$F(Fj]l7$F?$!3'Gx%\\?()yJFFiu7$FI$!3fxi(4^w#f9 Fht7$FS$!3?yuOiJ$f.%Fht7$Fhn$!3W4wrJ9c3%Fht7$Fjq$!3.)Q63Y4zX\"Fht7$F_r$!3%R5\\h*Q!H6#Fiu7$Fdr$!3[Rk)y\"GdlD Fhw7$Fir$\"3C_gB/hlUSFiu7$F_s$\"3HF\"eJ*Rd!>$Fht7$Fds$\"3Ue\\H7N^99F]t 7$Fis$\"3>cen\\#zqs$F]t7$F_t$\"3-ka')*3D'p()F]t7$Fdt$\"3;2&p5#4cQ^O#QPy\"R\"F07$F`v$\"31*yBA'))y=?F07$Fev$\"3=e*Gu0&GLHF07$Fjv$ \"3*p*HLgbh.UF07$F_w$\"3u'yU:\"HeFfn7$Fbz$\"3r>8:(R8*4sFfn7$Fgz$\"3F37$F`\\l$\"3BS-[')y9]@ F37$F\\\\n$\"34Y7J#=\\%QBF37$Fe\\l$\"3!)oG44N=SDF37$Fd\\n$\"3+aNF4BaYF F37$Fj\\l$\"3qG:U'>xl'HF37$F\\]n$\"3e<==+=4>KF37$F_]l$\"3`,++Sm8*[$F3F gbm-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F($\"+++++7!\"*;$!+++++?!#6$ \"+++++?Fjcq" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} }{PARA 3 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {XPPEDIT 18 0 "diff(y,t,t)^3-18*diff(y,t,t)*diff(y,t)+108*y" "6#,(*$-%%diffG6%%\"y G%\"tGF)\"\"$\"\"\"*(\"#=F+-F&6%F(F)F)F+-F&6$F(F)F+!\"\"*&\"$3\"F+F(F+ F+" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "ode := diff(y(t),t,t)^3-18*diff(y(t),t,t)*dif f(y(t),t)+108*y(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG,(*$)-%% diffG6$-%\"yG6#%\"tG-%\"$G6$F.\"\"#\"\"$\"\"\"F4*(\"#=F4F(F4-F)6$F+F.F 4!\"\"*&\"$3\"F4F+F4F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "e quations(Rosenfeld_Groebner([ode],R));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7#,(*$)-%%diffG6$-%\"yG6#%\"tG-%\"$G6$F.\"\"#\"\"$\"\"\"F4*(\" #=F4F(F4-F)6$F+F.F4!\"\"*&\"$3\"F4F+F4F47#,&*$)F7F3F4F2*&\"#FF4)F+F2F4 F97#F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 274 56 "Non-singular s olutions and first order singular solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 243 "ps1b:=plot(\{x^3/2\},x=- 7/10..7/10,-1/10..1/10, thickness=2): enps1:=plot(\{(x)^3,(x+1/3)^3+1/ 3*(x+1/3),(x+1/3)^3-1/24*(x+1/3),(x-1/3)^3+1/3*(x-1/3),(x-1/3)^3-1/24* (x-1/3)\},x=-7/10..7/10,-1/10..1/10):plots[display](\{ps1b,enps1\}, ti ckmarks=[4,4]);" }}{PARA 13 "" 1 "" {GLPLOT2D 355 262 262 {PLOTDATA 2 "6+-%'CURVESG6%7S7$$!3c**************p!#=$!3e***********\\r\"F*7$$!3dm mmT)R[p'F*$!3rlit#3V.]\"F*7$$!3[KLe>;KHkF*$!3#>RFe'y\")G8F*7$$!3cmm;4' =28'F*$!3czm0zq8_6F*7$$!3`mm;ki8IeF*$!3]Pzg'3f%3**!#>7$$!3+LLeMD)4`&F* $!3#yC,'pm7g%)FA7$$!3Vmm\"HtGOD&F*$!3gGsgk\"p,D(FA7$$!3h***\\i$\\Wm\\F *$!3*)f;hy5,DhFA7$$!3cmm\"H/R%pYF*$!3m)Hz=HV04&FA7$$!3n***\\7l&QtVF*$! 3l#o+A[yB=%FA7$$!3$HLL$[$e)oSF*$!3^5\"yPC@\"oLFA7$$!3Imm;M0j+QF*$!3RMV jjf'\\u#FA7$$!3%******\\ap')\\$F*$!3`jx7=iIT@FA7$$!3)******\\noa>$F*$! 3)*QH%ft\\9j\"FA7$$!3f*****\\XyK!HF*$!3R61C[/fB7FA7$$!3$om;HuTzj#F*$!3 '\\>*>&R9 QP\"Fhr7$$\"3e!***\\ioY/dFA$\"3eXea=jV\"G*Fhr7$$\"3gKLL3TU1')FA$\"3sVK $3B7u=$F]r7$$\"31*******)HWg6F*$\"3L\"3t^sCM\"yF]r7$$\"3x)***\\n$RPX\" F*$\"3>#RP\"pf8O:Fcp7$$\"3z***\\Pp=vt\"F*$\"3.Rjhd%fFi#Fcp7$$\"3U**** \\_sg_?F*$\"3KLy$*z)=SK%Fcp7$$\"3![mmmLGdL#F*$\"3lFIvG6VrjFcp7$$\"3&** ****\\_?!QEF*$\"3FUTB*G$>z\"*Fcp7$$\"3.KL3x@%>\"HF*$\"3\"\\#3LzrdM7FA7 $$\"3x*****\\*3T6KF*$\"3AY;54(*)fl\"FA7$$\"3`kmT?w=$\\$F*$\"3N$oy6gc78 #FA7$$\"3/++v)[Dxy$F*$\"3DWMV%>*4%FA7$$\"3OMLL`f^nYF*$\"38[!eGKcU3& FA7$$\"3FKL$eXWW'\\F*$\"3N>cX4EhR1(fm &=8F*7$$\"3(*)**\\iU4ep'F*$\"32v(o;/&*4]\"F*7$$\"3c**************pF*$ \"3e***********\\r\"F*-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fd[lFc[l-%*THI CKNESSG6#\"\"#-F$6$7S7$F($!3\\9[\"[\"[Jg5!#<7$F.$!3Ffo*zJyhD%F*7$Ffn$!3I5Xr2pTZPF*7$F[o$!3P#Hk!f6ZLLF*7$F`o$!3VN&*o0O %FA7$Fjr$!3K\\PNV1&pH$FA7$F_s$!3j^6T2v,NBFA7$Fes$!3tN>6\"R2I_\"FA7$Fjs $!3(>(\\N\"H&ey&*Fcp7$F_t$!3.)Rf#z*[c\"[Fcp7$Fdt$!36Q\\^e')[07Fcp7$Fit $\"3\\qp,cQF\">\"Fcp7$F^u$\"3R,e[BsG&e#Fcp7$Fcu$\"3%)R:#G\"ejNKFcp7$Fh u$\"3`]gs,`&Q;$Fcp7$F]v$\"3/\"Q2'R3)4c#Fcp7$Fbv$\"3[#Q(HV)p4o\"Fcp7$Fg v$\"3l=W/Gx(>1&F]r7$F\\w$!3MD\\x+ru>mF]r7$Faw$!3z%>NbG\"[*z\"Fcp7$Ffw$ !3(o?uM8XSo#Fcp7$F[x$!3g>O%4P`>@$Fcp7$F`x$!37*yw\\Y\">%=$Fcp7$Fex$!3') pm5\")oocCFcp7$Fjx$!3qH`%4MeY$))F]r7$F_y$\"3%p_LM!ovT9Fcp7$Fdy$\"3?De$ [*4Q+`Fcp7$Fiy$\"3(e_btG?:(**Fcp7$F^z$\"38ad*f!30P;FA7$Fcz$\"3$f<)z0jm +CFA7$Fhz$\"3N]=&=&=&=S$FAF\\[l-F$6$7S7$F($!3W9[\"[\"[\"yW\"F_\\l7$F.$ !3!f=ieA[FM\"F_\\l7$F3$!3I)f*\\F\\*eD\"F_\\l7$F8$!3X`*F*7$FM$!3&RZ <&=X+%[)F*7$FR$!3TQKN*f:Hz(F*7$FW$!3,AUaYrF*7$Ffn$!3K4&R$)yQK_'F*7$ F[o$!30\"zw$4wq3gF*7$F`o$!3J=9&p%zDmaF*7$Feo$!3\\%pgR)e=f\\F*7$Fjo$!3& =Kc)>6i/XF*7$F_p$!30=_d#R]&>TF*7$Fep$!38\"*3Vo;S%p$F*7$Fjp$!37B\"\\^i \\2O$F*7$F_q$!3nH'R>C9;+$F*7$Fdq$!3'3*3;/rM3FF*7$Fiq$!3)*zCfIrk6CF*7$F _r$!3!>@zL\"f-_@F*7$Fdr$!3Z^%*yoJR.>F*7$Fjr$!3lCT0\"z?Pp\"F*7$F_s$!31: O0'4$4'[\"F*7$Fes$!3hV\\?!3V\"*G\"F*7$Fjs$!3/+)pbMo=8\"F*7$F_t$!3/R4Id e:a(*FA7$Fdt$!3n(\\^Ll())o#)FA7$Fit$!3R2LF')*\\$HpFA7$F^u$!3i>*)3whxDd FA7$Fcu$!3()[G%=Af\"zWFA7$Fhu$!3i,uK2KjCMFA7$F]v$!3yi#*GZ\\K^BFA7$Fbv$ !3#ow2;&)>@T\"FA7$Fgv$!3YjbpkO*e1%Fcp7$F\\w$\"3KFuab5cK`Fcp7$Faw$\"3J! )*eUXAS_\"FA7$Ffw$\"3xxDSILV:DFA7$F[x$\"3eJ\"o>XjJf$FA7$F`x$\"3hDB]ycw %o%FA7$Fex$\"3oGVE@!)*4(eFA7$Fjx$\"3EWI:==eKrFA7$F_y$\"3J`L%G*e!)z$)FA 7$Fdy$\"3l'eej4J'G**FA7$Fiy$\"3%[bt.G\"fV6F*7$F^z$\"3TDeDQZ[=8F*7$Fcz$ \"3.nNUl^*4]\"F*7$Fhz$\"3f%=&=&=&=:B`c!>JqX\"Fcp7$FH$\"39)zl(eWv+#*F ]r7$FM$\"3k[#*)*\\c.\\CFcp7$FR$\"3?&)yj!)*>>=$Fcp7$FW$\"3&4oO\\b<&3KFc p7$Ffn$\"3X-B)3qqnm#Fcp7$F[o$\"3-hDFcp7$Fe p$!3s>\\:w%G!zJFcp7$Fjp$!3ZY.\\8'HvB$Fcp7$F_q$!3JUnA\"RB%3EFcp7$Fdq$!3 r(3<5:Q$)G\"Fcp7$Fiq$\"3-M7*Q0M$R6Fcp7$F_r$\"3YY%)G')z0]XFcp7$Fdr$\"3u I%)*oN\"[3%*Fcp7$Fjr$\"3:!Ri)=Wf<:FA7$F_s$\"3;tPWDwq%H#FA7$Fes$\"3y1bM )*Gt)G$FA7$Fjs$\"3-T8?W**eAVFA7$F_t$\"3T7PN]aZHcFA7$Fdt$\"3s5%e)HmL-sF A7$Fit$\"3')>*pCJta(*)FA7$F^u$\"3QhRUx+h#4\"F*7$Fcu$\"3&yH%pF'ezL\"F*7 $Fhu$\"3lZ<_))os&e\"F*7$F]v$\"3'3HB%HKS!)=F*7$Fbv$\"3P+8$*4Tlv@F*7$Fgv $\"3@Yf*z#plIDF*7$F\\w$\"3mM'[h;:o*GF*7$Faw$\"3N&*GtD5M9LF*7$Ffw$\"37( =!HcCPePF*7$F[x$\"3N@@'fY`FE%F*7$F`x$\"33IuB$3i#)y%F*7$Fex$\"3cdF;Xy`n `F*7$Fjx$\"3saMB-TO&)fF*7$F_y$\"3w#o*RbwX#f'F*7$Fdy$\"3+)y>>L5iL(F*7$F iy$\"38m,J'*HAY!)F*7$F^z$\"3)yE5SmH8&))F*7$Fcz$\"3:ZW2oB!)p'*F*7$Fhz$ \"3\\9[\"[\"[Jg5F_\\l-F][l6&F_[lF`[lF`[lFc[l-F$6$7S7$F($!3f%=&=&=&=:V.]\"F*7$F3$!3z#pCW,^(G8F*7$F8$!3Iv!)f\\tO^6F*7$F=$!38[U 4%f*=z)*FA7$FC$!3&)HSuBw)oQ)FA7$FH$!3aQ)\\Q?+\"4rFA7$FM$!3Sv&Qf%\\EzeF A7$FR$!3/5P(GT/Ap%FA7$FW$!3HHQpOWMzNFA7$Ffn$!3bn<;ONe*3\")f%Fcp7$Fjo$\"3Vy2 g0+ZT9FA7$F_p$\"3#)))=#>)*)f^BFA7$Fep$\"3a2X8@>)HZ$FA7$Fjp$\"3#\\'4lj) G%pWFA7$F_q$\"3MA)RuB\\#)p&FA7$Fdq$\"3;*G)RZaGioFA7$Fiq$\"3_A;&pg]tB)F A7$F_r$\"3aVjJ.%Qml*FA7$Fdr$\"3=%[J,a[n7\"F*7$Fjr$\"3>*[n^rLxG\"F*7$F_ s$\"3@x=B<%zoZ\"F*7$Fes$\"3H+3u[4.#p\"F*7$Fjs$\"3v$)Qw,l<'*=F*7$F_t$\" 39ry4\\%)oN@F*7$Fdt$\"3sSeB%y*R0CF*7$Fit$\"36\"\\fSf*p#p#F*7$F^u$\"3I6 -e(ezT*HF*7$Fcu$\"3](z\"QZjodLF*7$Fhu$\"3AZ TF*7$Fbv$\"3Z\\vLECj=H!4tIM\"F_\\l7$Fhz$\"3W9[\"[\"[\"yW\"F_\\l-F ][l6&F_[lFc[lFc[lF`[l-F$6$7S7$F($!3;************HMF*7$F.$!3UJDZlho+IF* 7$F3$!3%Qya;tNwl#F*7$F8$!37fL6eTF/BF*7$F=$!3](e@t\"=p\")>F*7$FC$!3c\\- #RLD?p\"F*7$FH$!3sX9#H$Q.]9F*7$FM$!3)>LAd@-]A\"F*7$FR$!3tfePe'3\"=5F*7 $FW$!3Hl8Skpvk$)FA7$Ffn$!3,@ib([Uit'FA7$F[o$!3xo'os#>$**[&FA7$F`o$!32F bDOCh#G%FA7$Feo$!3&z(e)=Z**GE$FA7$Fjo$!3yA7['*3=ZCFA7$F_p$!3*\\./e=uc$ =FA7$Fep$!3$>)\\Ofek_7FA7$Fjp$!310DJ]gB!o)Fcp7$F_q$!3#Ho:J\\\\mI&Fcp7$ Fdq$!3&G`(\\:/FqJFcp7$Fiq$!3Q8#*>yqo*e\"Fcp7$F_r$!3vaaX!p5Y!oF]r7$Fdr$ !3,%Qhn5F]r7$Fjr$!3J#=Dw()[8\"GFhr7$F_s$!3hbUl-yv*H$Fcs7$Fes$\"3'Q Q)R!zGwu#Fhr7$Fjs$\"37p\"4PE(Gc=F]r7$F_t$\"3X([m;YC[P'F]r7$Fdt$\"3F;Y. X\\oi:Fcp7$Fit$\"3P%yu#Q>FsIFcp7$F^u$\"32yEB:*=bC&Fcp7$Fcu$\"3kmc()fx. [')Fcp7$Fhu$\"3`01vDiGu7FA7$F]v$\"3XGo%ylQe$=FA7$Fbv$\"3$)\\;meV:pCFA7 $Fgv$\"3X#H.#=%z>J$FA7$F\\w$\"3pmtN-K^iUFA7$Faw$\"3]))o'))Q)>MaFA7$Ffw $\"3gb%\\2fI-x'FA7$F[x$\"3iq#f'ehW'Q)FA7$F`x$\"3j4;dk7&o,\"F*7$Fex$\"3 (Q7\"*=_ANA\"F*7$Fjx$\"3Oh!e^t9WX\"F*7$F_y$\"3*)4q'HPm1p\"F*7$Fdy$\"3/ HrF*7$Fiy$\"321j`hK%))G#F*7$F^z$\"3?Ry7%>Lrj#F*7$Fcz$\"38]vL$3 !*>+$F*7$Fhz$\"3;************HMF*-F][l6&F_[lF`[lFc[lF`[l-%*AXESTICKSG6 $\"\"%Fc[o-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;$!+++++q!#5$\"+++++qF a\\o;$!+++++5Fa\\o$\"+++++5Fa\\o" 1 2 0 1 10 0 2 9 1 4 2 1.000000 46.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 275 57 "Non-singular solutions and zero order singular solut ion " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "plot(\{seq( (x-1/2)^3+c/3*(x-1/2),c=-1..1), seq((x+1/ 2)^3+c/3*(x+1/2),c=-1..1)\},x=-1..1,-1/4..1/4);" }}{PARA 13 "" 1 "" {GLPLOT2D 355 264 264 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$!\"\"\"\"!$\"3ul mmmmmmT!#>7$$!3pmmm;p0k&*!#=$\"3V,3$Gc6jq&F-7$$!3vKL$3M+a#>sF-7$$!3:mmm\"4m(G$)F1$\"3p$4Qsb 'Q2uF-7$$!3\"QLL3i.9!zF1$\"3E')p5`B!*GsF-7$$!3\"ommT!R=0vF1$\"3,Y*)*\\ -t$ynF-7$$!3w****\\P8#\\4(F1$\"3RSZ6fRnjgF-7$$!3*pmmT?F1n'F1$\"3([],s3 '[-^F-7$$!3[++](y$pZiF1$\"3l$f@b$eukRF-7$$!33LLL$yaE\"eF1$\"3q.(pA#4=b EF-7$$!3hmmm\">s%HaF1$\"3W\"F-7$$!30+++]1aZTF1$!3eNSseSezF F-7$$!3umm;/#)[oPF1$!3)Rrkphl#=RF-7$$!3hLLL$=exJ$F1$!3@!f8rz398&F-7$$! 3*RLLLtIf$HF1$!3CJJsY&e3+'F-7$$!3]++]PYx\"\\#F1$!3NVYbbyx#y'F-7$$!3GML LL7i)4#F1$!3A2c7tA))GsF-7$$!3d****\\P'psm\"F1$!3_/b&Q/2uS(F-7$$!3')*** *\\74_c7F1$!3z\"[]ZT)GKsF-7$$!3)3LLL3x%z#)F-$!3S33!RFe\\k'F-7$$!3JMLL3 s$QM%F-$!3,S4$QOhw$3#F-7$$\"3r*)***\\PQ#\\\")F-$\"3%H:D^w(>\"z#!#?7$$\"3GKL Le\"*[H7F1$\"3Pj5l_^_4MF-7$$\"3I*******pvxl\"F1$\"3_7[.pdl=tF-7$$\"3#z ****\\_qn2#F1$\"3'*4$))yOr^=\"F17$$\"3V)***\\i&p@[#F1$\"3:44@#4wYp\"F1 7$$\"3B)****\\2'HKHF1$\"3?%ycgy1qM#F17$$\"3ElmmmZvOLF1$\"3)QiWXp\\_,$F 17$$\"3h******\\2goPF1$\"3ND)prJm\">QF17$$\"3UKL$eR<*fTF1$\"3.m*)pM#RA j%F17$$\"3n******\\)Hxe%F1$\"3^;=5\"))ovh&F17$$\"3dkm;H!o-*\\F1$\"3,eS g[JuSmF17$$\"3y)***\\7k.6aF1$\"3O\\$Qi\\VT\"yF17$$\"3#emmmT9C#eF1$\"30 0tvYRCo!*F17$$\"32****\\i!*3`iF1$\"3&3J]nN)*)\\5!#<7$$\"3%QLLL$*zym'F1 $\"3:?kf+\"H&*>\"F_x7$$\"3xKLL3N1#4(F1$\"3z/]6d&3]O\"F_x7$$\"3Mmm;HYt7 vF1$\"3Uade'G4?a\"F_x7$$\"3Z*******p(G**yF1$\"3LS#R9;djr\"F_x7$$\"3]mm mT6KU$)F1$\"3**yl/iIUI>F_x7$$\"3fKLLLbdQ()F1$\"3H6#H\\U#=N@F_x7$$\"3[+ +]i`1h\"*F1$\"3!)*RubmkxO#F_x7$$\"3W++]P?Wl&*F1$\"3$[kdlSuXg#F_x7$$\" \"\"F*$\"3+++++++vGF_x-%'COLOURG6&%$RGBG$\"#5F)$F*F*Fc[l-F$6$7S7$F($!3 Immmmmm;HF17$F/$!3qI!G[Xt?Z#F17$F5$!3O&Q'G([^x7#F17$F:$!3%Ha-T*3_$y\"F 17$F?$!3kM1sQ(Q%y9F17$FD$!3yc[CX)y8@\"F17$FI$!34G)yF+`G#**F-7$FN$!3:c_ )3HoC!zF-7$FS$!3))RHC2'G].'F-7$FX$!31T<\"yu7KN%F-7$Fgn$!3)>&eGLw^iFF-7 $F\\o$!3#**H;1V&\\R9F-7$Fao$\"3&*oAg'o+bL'Feo7$Fgo$\"3UNOx%*HQe9F-7$F \\p$\"3U(H4YFyM!HF-7$Fap$\"3CwT#>-8=H%F-7$Ffp$\"3`]3LZm`$3'F-7$F[q$\"3 MSY0JKgfxF-7$F`q$\"38=?6hXsQ**F-7$Feq$\"3d\\&)z$Gk8@\"F17$Fjq$\"3T;6GP &z5[\"F17$F_r$\"3t^\\_$)QUs$3&)Q2\"F_x7$Fgv$\"3_\"=5\"y)Q4?\"F_x7$F\\w$\"3 #p^rM=#4I8F_x7$Faw$\"3c,0zVM[v9F_x7$Ffw$\"33v,7*o=$G;F_x7$F[x$\"3$4J]U H/,!=F_x7$Fax$\"3qU'=Gw(Qx>F_x7$Ffx$\"3)pAPV7Y6<#F_x7$F[y$\"3^lo>&f\"> wBF_x7$F`y$\"3btDxu'4jd#F_x7$Fey$\"3Y!pd\"Q6\"*>GF_x7$Fjy$\"3kmZ[gu3^I F_x7$F_z$\"3!f1T(*oN=J$F_x7$Fdz$\"3$4JCd(QgvNF_x7$Fiz$\"3+++++++vQF_x- F^[l6&F`[lFc[lFa[lFc[l-F$6$7S7$F($!3+++++++vGF_x7$F/$!3+Xb>#4RPg#F_x7$ F5$!3K.3g>^C\"Q#F_x7$F:$!3B!z+d3Sc9#F_x7$F?$!3]y^QRJkB>F_x7$FD$!3PS^^= IM<nhO\"F_x7$FS$!3:/PqC)f0?\" F_x7$FX$!3;LC=Q&H![5F_x7$Fgn$!3Ig?]j_BP!*F17$F\\o$!3qV$3$**=0oyF17$Fao $!3kM\"R.N*fhmF17$Fgo$!3?0M(38.Dc&F17$F\\p$!3$yzfb5`_g%F17$Fap$!3'>8Y= BW*=QF17$Ffp$!31<1IbL1#)HF17$F[q$!3hU$=EIfEN#F17$F`q$!3c3'[aqEwq\"F17$ Feq$!3=&[y#=#=3@\"F17$Fjq$!3\\6.\"H%yV8uF-7$F_r$!3i[m1#\\*[NOF-7$Fdr$! 3[9tqP;<\"o$F]t7$Fir$\"3'pNF(=V]l?F-7$F^s$\"3Cp*zn#zQDTF-7$Fds$\"3jEQM c(*3#p&F-7$Fis$\"32t'oq_],i'F-7$F_t$\"3Bvng9x#z?(F-7$Fdt$\"3S\"oOyGGtS (F-7$Fit$\"3WH,pr&>hC(F-7$F^u$\"3YD\"QW/(f'z'F-7$Fcu$\"3w!**>&z.K3gF-7 $Fhu$\"3UM9;gJ.%3&F-7$F]v$\"3:-*HjmTz\"RF-7$Fbv$\"3,fE&)fu)4u#F-7$Fgv$ \"3)*H%3YhEsO\"F-7$F\\w$\"33Sk]g5)RC$F`s7$Faw$!3#)Qd[wo!)RF-7$Fax$!32kE2B]i&4&F-7$Ffx$!3&[3mQP0z0'F -7$F[y$!38P7$H$GG*y'F-7$F`y$!3w]k&[-*=FsF-7$Fey$!3/HY5alK2uF-7$Fjy$!3/ _81H7`OsF-7$F_z$!3k.&=s$eblmF-7$Fdz$!3Py'=#H,F-dF-7$Fiz$!3ulmmmmmmTF-- F^[l6&F`[lFa[lFa[lFc[l-F$6$7S7$F($!3+++++++vLF_x7$F/$!3[LW3cw?*3$F_x7$ F5$!3$4eG'e*pS&GF_x7$F:$!3gXjvLdC/EF_x7$F?$!36M2WU`$zO#F_x7$FD$!3]^i(e !)*QZ@F_x7$FI$!3>r(\\H'fbb>F_x7$FN$!3@O#=_BJ$p2Sva@3v)F17$F\\p$!3yjkAALVawF17$Fap$! 3x>]BLOxTnF17$Ffp$!3*ys6kT\\Yv&F17$F[q$!3=(yiqapz*\\F17$F`q$!3k3'[z\" \\)[?%F17$Feq$!3=ji0'HDqd$F17$Fjq$!3tI5z'*pwjHF17$F_r$!31)**R+#f0\\CF1 7$Fdr$!3*R=)[_4Yz>F17$Fir$!3-U]S33\"\\g\"F17$F^s$!3%=*3@r%=uD\"F17$Fds $!3zm]adUlP&*F-7$Fis$!3*)H8$zM.,L(F-7$F_t$!30]ahdMt$F-7$Fit$!3nw)4$y'yz\\#F-7$F^u$!3Wz=c!3rhf\"F-7$Fcu$!39DoYr.ES))F] t7$Fhu$!37lM;wh<,YF]t7$F]v$!3ypwO.]An=F]t7$Fbv$!3`al&pt)yGfF`s7$Fgv$!3 o!)f!\\s=s+(Feo7$F\\w$!3_R$)3R=F<#*!#F7$Faw$\"3U1$eU^)\\WpFeo7$Ffw$\"3 $y/zr:FDc&F`s7$F[x$\"3Aisf$zSw'>F]t7$Fax$\"3?f60ZvuRYF]t7$Ffx$\"3V#*o6 RlRc\"*F]t7$F[y$\"3W$)4Hk#*\\'e\"F-7$F`y$\"3.\")oZ3L5PCF-7$Fey$\"3$\\f <\"=RuLPF-7$Fjy$\"3X'3$Q:sQD_F-7$F_z$\"3b'\\\"yP?m/sF-7$Fdz$\"3J?8y&** pe^*F-7$Fiz$\"3+++++++]7F1-F^[l6&F`[lFc[lFc[lFa[l-F$6$7S7$F($!3+++++++ ]7F17$F/$!3T^Zs#\\6s]*F-7$F5$!3`6%>MIk$GtF-7$F:$!3(*4K!)ou(zI&F-7$F?$! 3.FT)\\T+&)o$F-7$FD$!3$=zq'\\IWUCF-7$FI$!3RT**))))*RAd\"F-7$FN$!3OvD&) e;(R>*F]t7$FS$!3%)pr?+EriYF]t7$FX$!3D]2XhXLU>F]t7$Fgn$!3!\\G23bNoO&F`s 7$F\\o$!3clt2%Re9#zFeo7$Fao$\"3&QX?&3>/mo!#H7$Fgo$\"3of$op5GQB)Feo7$F \\p$\"3/eJE%z5Z>'F`s7$Fap$\"3=2tzCqtn=F]t7$Ffp$\"3L.j3^#R1w%F]t7$F[q$ \"3.Uvl@Ms$z)F]t7$F`q$\"3u(oyFNtzd\"F-7$Feq$\"3EW*HCG!QUCF-7$Fjq$\"3[I yZkTp,PF-7$F_r$\"3u<&\\-@vfC&F-7$Fdr$\"3D89K)p#)=E(F-7$Fir$\"3H9Ysm&fp ^*F-7$F^s$\"3=@.Zk2hU7F17$Fds$\"3/tsTNF)>g\"F17$Fis$\"3_%e%[B*>i'>F17$ F_t$\"3#o@w8!*[uT#F17$Fdt$\"3TZ,dVT7^HF17$Fit$\"3KU;Aw[4WNF17$F^u$\"30 UU/8Et)=%F17$Fcu$\"3P]Msxa5\"*\\F17$Fhu$\"3gy,5]z;%z&F17$F]v$\"3y!\\OQ BL?u'F17$Fbv$\"35w+JL]a&o(F17$Fgv$\"3H:=5J)yM\"))F17$F\\w$\"3]7'f;\\K3 (**F17$Faw$\"3>orq'*)[%G6F_x7$Ffw$\"3u_z*=arvE\"F_x7$F[x$\"3*3J+bK,]U \"F_x7$Fax$\"3VJvqJ%e%)e\"F_x7$Ffx$\"3*e6E2Mx!oF_x7$F`y$\"302f5=MLY@F_x7$Fey$\"3$[8-,5n^P#F_x7$Fjy$\"3(*))pqU\\8$f# F_x7$F_z$\"3&Gtdw<+)RGF_x7$Fdz$\"3)y(49T\"*3!4$F_x7$Fiz$\"3+++++++vLF_ x-F^[l6&F`[lFa[lFc[lFa[l-F$6$7S7$F($!3+++++++vQF_x7$F/$!3(>Kt*>inuNF_x 7$F5$!3aejl(z%*oK$F_x7$F:$!3(4!>\"=Q^G1$F_x7$F?$!3]*G'\\XvA7GF_x7$FD$! 3TitB$fOud#F_x7$FI$!3I$*>UEaRsBF_x7$FN$!3`p:!)z_\\s@F_x7$FS$!3]:[JQ;gy >F_x7$FX$!3ULCo!zvyz\"F_x7$Fgn$!3FGCdyrcC;F_x7$F\\o$!3bX>/OL5#[\"F_x7$ Fao$!37!e+<*)*pK8F_x7$Fgo$!3(Rn?k*R\"R>\"F_x7$F\\p$!338$*)QNh.2\"F_x7$ Fap$!3d2RiMIgk'*F17$Ffp$!3sQG_xaBF&)F17$F[q$!3ZJs]\"zzKk(F17$F`q$!3**3 '[/8V@q'F17$Feq$!3!4/MQPKK%fF17$Fjq$!3eI5H4->'=&F17$F_r$!3`JL(3*GcMXF1 7$Fdr$!3$[H*f)=5@#RF17$Fir$!3u>Go[?P;MF17$F^s$!3)3y*4NdPFHF17$Fds$!3-' RVr#)RnZ#F17$Fis$!3ULJHAd.G@F17$F_t$!3nnP)H7\")Gz\"F17$Fdt$!3dl'\\Xq;u [\"F17$Fit$!3N))4$Gp2UA\"F17$F^u$!3.&)=c0#R*))**F-7$Fcu$!3SaL\"QXsjx(F -7$Fhu$!3lGTR&RoU+'F-7$F]v$!3jNM+nmQ\"H%F-7$Fbv$!3^\"z\"fMKcfGF-7$Fgv$ !3xS#e?0T7Q\"F-7$F\\w$!3ZJ.0&\\**RC$F`s7$Faw$\"3\"\\D9N(e1x8F-7$Ffw$\" 3#png/m1qz#F-7$F[x$\"3!4RE5iFPP%F-7$Fax$\"3([*G[KXdBgF-7$Ffx$\"32l%*o \"o%=*)yF-7$F[y$\"341K^h8Gi**F-7$F`y$\"39@5=k&R,@\"F17$Fey$\"3h\")R.R9 [([\"F17$Fjy$\"3j_F)fcI(o " 0 "" {MPLTEXT 1 0 33 "ode := diff(y(t),t,t)^3-216*y(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG,&*$)-%%diffG6$-%\"yG6#%\"tG-%\"$G6$F.\"\"# \"\"$\"\"\"F4*&\"$;#F4F+F4!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "equations(Rosenfeld_Groebner([ode],R));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7#,&*$)-%%diffG6$-%\"yG6#%\"tG-%\"$G6$F.\"\"#\"\"$\" \"\"F4*&\"$;#F4F+F4!\"\"7#F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "plot( \{seq((x+a)^3,a=-1..1), seq(-(x+a)^3,a=-1..1)\}, x=-2..2, \+ -1..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 355 258 258 {PLOTDATA 2 "6*-%'CU RVESG6$7S7$$!\"#\"\"!$!#FF*7$$!3MLLL$Q6G\">!#<$!3zYv'[7m8Z#!#;7$$!3bmm ;M!\\p$=F0$!3ukG!p'fD$G#F37$$!3LLLL))Qj^G;8,\"F37$$!3KLLLQW*e3\"F0$!3!)=6*Q5Mc2*F07$$!3x++++()>'** *!#=$!3*Q(z$pIRa*zF07$$!3E++++0\"*H\"*Fco$!3`d?.Qsl+qF07$$!35++++83&H) Fco$!3-r6ydmaBhF07$$!3\\LLL3k(p`(Fco$!3#e,)e1*=MR&F07$$!3Anmmmj^NmFco$ !3J#QHJ`>Pg%F07$$!3(zmmmYh=(eFco$!3uH-lPcP)*RF07$$!3+,++v#\\N)\\Fco$!3 \"p))eV$z!RO$F07$$!3commmCC(>%Fco$!3b5]%oB?;'GF07$$!39*****\\FRXL$Fco$ !3eCGV(f85P#F07$$!3t*****\\#=/8DFco$!3X)*>.OZCf>F07$$!3=mmm;a*el\"Fco$ !3>Z0*>woNe\"F07$$!3iomm;Wn(o)!#>$!3iLSsY'GRG\"F07$$!3IqLLL$eV(>!#?$!3 Z8(opxMf+\"F07$$\"3)Qjmm\"f`@')F[s$!3W`g.1M7IwFco7$$\"3%z****\\nZ)H;Fc o$!3\"R1X$yE3keFco7$$\"3bkmm;$y*eCFco$!3/gB41MN)G%Fco7$$\"3f)******R^b J$Fco$!3F!*R1.bt')HFco7$$\"3&e*****\\5a`TFco$!3M,zkUHQ)*>Fco7$$\"3'o** **\\7RV'\\Fco$!3b.&\\W'o$pF\"Fco7$$\"3X'*****\\@fkeFco$!36gM<(H3A2(F[s 7$$\"3_ILLL&4Nn'Fco$!34s2$4%4%4o$F[s7$$\"3A*******\\,s`(Fco$!3#e8%p-+y $\\\"F[s7$$\"3%[mm;zM)>$)Fco$!3jV_c*)4.VZFas7$$\"3L*******pfa<*Fco$!3b %yC*z\\x0c!#@7$$\"38HLLeg`!)**Fco$!3ir1Fru\"QP(!#E7$$\"3w****\\#G2A3\" F0$\"38XmS6))fbbF[w7$$\"3;LLL$)G[k6F0$\"3EQKuD<-]WFas7$$\"3\")****\\7y h]7F0$\"3y4y([j7Td\"F[s7$$\"3xmmm')fdL8F0$\"3OF4kP!)z6PF[s7$$\"3bmmm,F T=9F0$\"3%R^$HJs6DtF[s7$$\"3FLL$e#pa-:F0$\"3v'yK9T*>p7Fco7$$\"3*)***** *Rv&)z:F0$\"3#[]\"yYEo\\>Fco7$$\"3ILLLGUYo;F0$\"3%f2%\\M^*p)HFco7$$\"3 _mmm1^rZD-jHPw&Fco7$$\"34+ +]2%)38>F0$\"3Dc]i'*fp7wFco7$$\"\"#F*$\"\"\"F*-%'COLOURG6&%$RGBG$\"#5! \"\"$F*F*Fe[l-F$6$7S7$F($Fd[lF*7$F.$!38,)zT>pdg(Fco7$F5$!3AHbtU9pieFco 7$F:$!3)zcU](>QYUFco7$F?$!3j,t)>L+3&HFco7$FD$!3YLmtRW&R&>Fco7$FI$!36`> 6\"*>zd7Fco7$FN$!3Hg?3Ft'*=%\\\"F [s7$Ffp$\"3mU!p3S6&3QF[s7$F[q$\"3jLgKP(y\\.(F[s7$F`q$\"3>]HA#oyBE\"Fco 7$Feq$\"3TbR%fA/R&>Fco7$Fjq$\"3QkAeJ`NhHFco7$F_r$\"3>9'*>o,y'>%Fco7$Fd r$\"3gIrleh]4eFco7$Fir$\"3W\"pzLlnNh(Fco7$F_s$\"3UpDw:h)3%**Fco7$Fes$ \"3V=QL)='e\"G\"F07$Fjs$\"3inwyQf(Hd\"F07$F_t$\"3Yt45@\"fR$>F07$Fdt$\" 3$z6c[J**3O#F07$Fit$\"3'QJx4!fFNGF07$F^u$\"3k$RN/4')4N$F07$Fcu$\"3Hg(y @Q%)G*RF07$Fhu$\"3)G9!3gVLNYF07$F]v$\"3i#>pqeEOR&F07$Fbv$\"3)31[m-O%[h F07$Fgv$\"3B_9)[1$y]qF07$F]w$\"3+!pFL*fmwzF07$Fcw$\"3aXtlt6fF!*F07$Fhw $\"3>i$=NBdS,\"F37$F]x$\"3r[-Sg5+S6F37$Fbx$\"39DgOXnwq7F37$Fgx$\"3r#*3 es=Y99F37$F\\y$\"3)z/8FN!Gn:F37$Fay$\"3kDZ[Mn1<F37$F[z$\"3<\"flT&z]u?F37$F`z$\"3G'=E@9S=F#F37$Fez$\"3I#y7HJr?Z#F37$F jz$\"#FF*-F_[l6&Fa[lFe[lFb[lFe[l-F$6$7S7$F(Fjdl7$F.$\"3zYv'[7m8Z#F37$F 5$\"3ukG!p'fD$G#F37$F:$\"3[w]+(e'R$3#F37$F?$\"3H(e_RF[V*=F37$FD$\"3?,5 qW=\"zr\"F37$FI$\"3&p\")f.xWWc\"F37$FN$\"3(*)eu\"))\\Y:9F37$FS$\"3'yS2 _ek;F\"F37$FX$\"3jYfaJ@OQ6F37$Fgn$\"3o`!p>G;8,\"F37$F\\o$\"3!)=6*Q5Mc2 *F07$Fao$\"3*Q(z$pIRa*zF07$Fgo$\"3`d?.Qsl+qF07$F\\p$\"3-r6ydmaBhF07$Fa p$\"3#e,)e1*=MR&F07$Ffp$\"3J#QHJ`>Pg%F07$F[q$\"3uH-lPcP)*RF07$F`q$\"3 \"p))eV$z!RO$F07$Feq$\"3b5]%oB?;'GF07$Fjq$\"3eCGV(f85P#F07$F_r$\"3X)*> .OZCf>F07$Fdr$\"3>Z0*>woNe\"F07$Fir$\"3iLSsY'GRG\"F07$F_s$\"3Z8(opxMf+ \"F07$Fes$\"3W`g.1M7IwFco7$Fjs$\"3\"R1X$yE3keFco7$F_t$\"3/gB41MN)G%Fco 7$Fdt$\"3F!*R1.bt')HFco7$Fit$\"3M,zkUHQ)*>Fco7$F^u$\"3b.&\\W'o$pF\"Fco 7$Fcu$\"36gM<(H3A2(F[s7$Fhu$\"34s2$4%4%4o$F[s7$F]v$\"3#e8%p-+y$\\\"F[s 7$Fbv$\"3jV_c*)4.VZFas7$Fgv$\"3b%yC*z\\x0cF[w7$F]w$\"3ir1Fru\"QP(Faw7$ Fcw$!38XmS6))fbbF[w7$Fhw$!3EQKuD<-]WFas7$F]x$!3y4y([j7Td\"F[s7$Fbx$!3O F4kP!)z6PF[s7$Fgx$!3%R^$HJs6DtF[s7$F\\y$!3v'yK9T*>p7Fco7$Fay$!3#[]\"yY Eo\\>Fco7$Ffy$!3%f2%\\M^*p)HFco7$F[z$!3;pkIs(4.=%Fco7$F`z$!3C(>D-jHPw& Fco7$Fez$!3Dc]i'*fp7wFco7$FjzFj[l-F_[l6&Fa[lFb[lFb[lFe[l-F$6$7S7$F($\" \")F*7$F.$\"3w?n/M\\o)*pF07$F5$\"3Z+6:/tc)>'F07$F:$\"3$R^x(o.Su`F07$F? $\"3V-BOJ<-AYF07$FD$\"32P=*Hx9k%RF07$FI$\"3Z%oa)3J+#Q$F07$FN$\"3'\\N3] -[r&GF07$FS$\"3fly$RX(fuBF07$FX$\"3$z`vMsj4&>F07$Fgn$\"3G\"flSFN6d\"F0 7$F\\o$\"3bA8G&f]/G\"F07$Fao$\"3sD_TM/g))**Fco7$Fgo$\"3m+v\"*e6E5wFco7 $F\\p$\"3V-)Gda5xq&Fco7$Fap$\"3h&=[p#eX\"G%Fco7$Ffp$\"3tcMg&*Hi@HFco7$ F[q$\"3pU[Q^[aC?Fco7$F`q$\"3UdHo0DqP7Fco7$Feq$\"3o!pIDr;UR(F[s7$Fjq$\" 3MR*HP'\\s2PF[s7$F_r$\"3ha=g573(e\"F[s7$Fdr$\"3^9sTi![/a%Fas7$Fir$\"3H a6KI>3dlF[w7$F_s$\"3!HpZ66Gip(Faw7$Fes$!351\\2^OY3kF[w7$Fjs$!3hd6ewH`H VFas7$F_t$!3FKpeC&Ro[\"F[s7$Fdt$!3<*)f[FiuWOF[s7$Fit$!3A;jiN)[c;(F[s7$ F^u$!3FEA&\\kWMA\"Fco7$Fcu$!3$)Q^.Y].:Xn7F07$Fhw$!3J$oa%GD1z:F07$F]x$!3%GjvwUAg&>F0 7$Fbx$!3CI$=qvk;P#F07$Fgx$!3gJ41Wop`GF07$F\\y$!3&*zojc4A#R$F07$Fay$!3! [qiZ=XK%RF07$Ffy$!31WK8saiWYF07$F[z$!3-!GV%R*4%Q`F07$F`z$!3#4>UY(zu]hF 07$Fez$!3ik^*=9E<+(F07$Fjz$!\")F*-F_[l6&Fa[lFe[lFe[lFb[l-F$6$7S7$F(F\\ [l7$F.$\"38,)zT>pdg(Fco7$F5$\"3AHbtU9pieFco7$F:$\"3)zcU](>QYUFco7$F?$ \"3j,t)>L+3&HFco7$FD$\"3YLmtRW&R&>Fco7$FI$\"36`>6\"*>zd7Fco7$FN$\"3Hg? 3Ft'*=%\\\"F[s7$Ffp$!3mU!p3S6&3Q F[s7$F[q$!3jLgKP(y\\.(F[s7$F`q$!3>]HA#oyBE\"Fco7$Feq$!3TbR%fA/R&>Fco7$ Fjq$!3QkAeJ`NhHFco7$F_r$!3>9'*>o,y'>%Fco7$Fdr$!3gIrleh]4eFco7$Fir$!3W \"pzLlnNh(Fco7$F_s$!3UpDw:h)3%**Fco7$Fes$!3V=QL)='e\"G\"F07$Fjs$!3inwy Qf(Hd\"F07$F_t$!3Yt45@\"fR$>F07$Fdt$!3$z6c[J**3O#F07$Fit$!3'QJx4!fFNGF 07$F^u$!3k$RN/4')4N$F07$Fcu$!3Hg(y@Q%)G*RF07$Fhu$!3)G9!3gVLNYF07$F]v$! 3i#>pqeEOR&F07$Fbv$!3)31[m-O%[hF07$Fgv$!3B_9)[1$y]qF07$F]w$!3+!pFL*fmw zF07$Fcw$!3aXtlt6fF!*F07$Fhw$!3>i$=NBdS,\"F37$F]x$!3r[-Sg5+S6F37$Fbx$! 39DgOXnwq7F37$Fgx$!3r#*3es=Y99F37$F\\y$!3)z/8FN!Gn:F37$Fay$!3kDZ[Mn1<< F37$Ffy$!3(yq\"3!oL,!>F37$F[z$!3<\"flT&z]u?F37$F`z$!3G'=E@9S=F#F37$Fez $!3I#y7HJr?Z#F37$FjzF+-F_[l6&Fa[lFb[lFe[lFb[l-F$6$7S7$F(Ffgm7$F.$!3w?n /M\\o)*pF07$F5$!3Z+6:/tc)>'F07$F:$!3$R^x(o.Su`F07$F?$!3V-BOJ<-AYF07$FD $!32P=*Hx9k%RF07$FI$!3Z%oa)3J+#Q$F07$FN$!3'\\N3]-[r&GF07$FS$!3fly$RX(f uBF07$FX$!3$z`vMsj4&>F07$Fgn$!3G\"flSFN6d\"F07$F\\o$!3bA8G&f]/G\"F07$F ao$!3sD_TM/g))**Fco7$Fgo$!3m+v\"*e6E5wFco7$F\\p$!3V-)Gda5xq&Fco7$Fap$! 3h&=[p#eX\"G%Fco7$Ffp$!3tcMg&*Hi@HFco7$F[q$!3pU[Q^[aC?Fco7$F`q$!3UdHo0 DqP7Fco7$Feq$!3o!pIDr;UR(F[s7$Fjq$!3MR*HP'\\s2PF[s7$F_r$!3ha=g573(e\"F [s7$Fdr$!3^9sTi![/a%Fas7$Fir$!3Ha6KI>3dlF[w7$F_s$!3!HpZ66Gip(Faw7$Fes$ \"351\\2^OY3kF[w7$Fjs$\"3hd6ewH`HVFas7$F_t$\"3FKpeC&Ro[\"F[s7$Fdt$\"3< *)f[FiuWOF[s7$Fit$\"3A;jiN)[c;(F[s7$F^u$\"3FEA&\\kWMA\"Fco7$Fcu$\"3$)Q ^.Y].:X n7F07$Fhw$\"3J$oa%GD1z:F07$F]x$\"3%GjvwUAg&>F07$Fbx$\"3CI$=qvk;P#F07$F gx$\"3gJ41Wop`GF07$F\\y$\"3&*zojc4A#R$F07$Fay$\"3![qiZ=XK%RF07$Ffy$\"3 1WK8saiWYF07$F[z$\"3-!GV%R*4%Q`F07$F`z$\"3#4>UY(zu]hF07$Fez$\"3ik^*=9E <+(F07$FjzFf^m-F_[l6&Fa[lFe[lFb[lFb[l-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VI EWG6$;F(Fjz;Fj[lF\\[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 0 {PARA 4 "" 0 "" {XPPEDIT 18 0 "diff(y,t)^2*diff(y,t,t)-8*y" "6 #,&*&-%%diffG6$%\"yG%\"tG\"\"#-F&6%F(F)F)\"\"\"F-*&\"\")F-F(F-!\"\"" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "ode := diff(y(t),t )^2*diff(y(t),t,t)-8*y(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG, &*&)-%%diffG6$-%\"yG6#%\"tGF.\"\"#\"\"\"-F)6$F+-%\"$G6$F.F/F0F0*&\"\") F0F+F0!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "equations(Ro senfeld_Groebner([ode],R));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7#,&* &)-%%diffG6$-%\"yG6#%\"tGF.\"\"#\"\"\"-F)6$F+-%\"$G6$F.F/F0F0*&\"\")F0 F+F0!\"\"7#F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "plot( \{se q((x+a)^2,a=-1..1), seq(-(x+a)^2,a=-1..1)\}, x=-2..2, -1..1);" }} {PARA 13 "" 1 "" {GLPLOT2D 355 235 235 {PLOTDATA 2 "6*-%'CURVESG6$7S7$ $!\"#\"\"!$!\"\"F*7$$!3MLLL$Q6G\">!#<$!3dO\"Ha@YAL)!#=7$$!3bmm;M!\\p$= F0$!3kf^#z&o$[+(F37$$!3LLLL))Qj^@xMFKWF37$$!3wmm;C2G!e\"F0$!3v6RR)=dsO$F37$$!3OLL$3yO5]\"F0$!3>(Hy uby.^#F37$$!3&*****\\nU)*=9F0$!3?:^7k\"yav\"F37$$!3SLL$3WDTL\"F0$!3#Qp ?@5)R;6F37$$!35++]d(Q&\\7F0$!3G&Qk%\\\"fpA'!#>7$$!3immmc4`i6F0$!3E'=) \\(=J;k#Ffn7$$!3KLLLQW*e3\"F0$!3l+))fOX&yP(!#?7$$!3x++++()>'***F3$!3\" [T***o\"))\\W\"!#C7$$!3E++++0\"*H\"*F3$!3\"e\\-,Qd0d(Fao7$$!35++++83&H )F3$!3l*o4OxZn!HFfn7$$!3\\LLL3k(p`(F3$!3mll58_[mgFfn7$$!3Anmmmj^NmF3$! 3S'y'*=,v>8\"F37$$!3(zmmmYh=(eF3$!3ZZ\"R]x_Tq\"F37$$!3+,++v#\\N)\\F3$! 3s,`j(yxk^#F37$$!3commmCC(>%F3$!3%Gnl!*\\*>nLF37$$!39*****\\FRXL$F3$!3 *Qv^wmOGW%F37$$!3t*****\\#=/8DF3$!3-M*>9Faag&F37$$!3=mmm;a*el\"F3$!3bM /wH\"3C'pF37$$!3iomm;Wn(o)Ffn$!3)3nOM!3%zL)F37$$!3IqLLL$eV(>Fao$!3<<;C 9=bg**F37$$\"3)Qjmm\"f`@')Ffn$!3!*p&*[1Q')z6F07$$\"3%z****\\nZ)H;F3$!3 jGqVpN`_8F07$$\"3bkmm;$y*eCF3$!37,<&pShAb\"F07$$\"3f)******R^bJ$F3$!3A >/'34RIx\"F07$$\"3&e*****\\5a`TF3$!3*)\\.aUsA.?F07$$\"3'o****\\7RV'\\F 3$!3]c+[aWJRAF07$$\"3X'*****\\@fkeF3$!3*\\Te3%G&o^#F07$$\"3_ILLL&4Nn'F 3$!3gu:e,#f+y#F07$$\"3A*******\\,s`(F3$!3-Ag^kV`vIF07$$\"3%[mm;zM)>$)F 3$!3'RgRzYjhN$F07$$\"3L*******pfa<*F3$!3oSK1ZD)pn$F07$$\"38HLLeg`!)**F 3$!3$Qe$y6#=A*RF07$$\"3w****\\#G2A3\"F0$!3AMgHnreNVF07$$\"3;LLL$)G[k6F 0$!3VzHC_h)\\o%F07$$\"3\")****\\7yh]7F0$!31%GUz`!Gl]F07$$\"3xmmm')fdL8 F0$!3Y2ta&)odXaF07$$\"3bmmm,FT=9F0$!3?lEe&**>([eF07$$\"3FLL$e#pa-:F0$! 3amy*f6TFE'F07$$\"3*)******Rv&)z:F0$!3K^[pE\\mbmF07$$\"3ILLLGUYo;F0$!3 C7Y*yL,27(F07$$\"3_mmm1^rZF0$!3$f)3\"*pS3'[)F07$$\"\"#F*$!\"*F*-%'COL OURG6&%$RGBG$\"#5F,$F*F*Fb[l-F$6$7S7$F($\"\"\"F*7$F.$\"3dO\"Ha@YAL)F37 $F5$\"3kf^#z&o$[+(F37$F:$\"3de3\"4-N&\\cF37$F?$\"3z3>@xMFKWF37$FD$\"3v 6RR)=dsO$F37$FI$\"3>(Hyuby.^#F37$FN$\"3?:^7k\"yav\"F37$FS$\"3#Qp?@5)R; 6F37$FX$\"3G&Qk%\\\"fpA'Ffn7$Fhn$\"3E'=)\\(=J;k#Ffn7$F]o$\"3l+))fOX&yP (Fao7$Fco$\"3\"[T***o\"))\\W\"Fgo7$Fio$\"3\"e\\-,Qd0d(Fao7$F^p$\"3l*o4 OxZn!HFfn7$Fcp$\"3mll58_[mgFfn7$Fhp$\"3S'y'*=,v>8\"F37$F]q$\"3ZZ\"R]x_ Tq\"F37$Fbq$\"3s,`j(yxk^#F37$Fgq$\"3%Gnl!*\\*>nLF37$F\\r$\"3*Qv^wmOGW% F37$Far$\"3-M*>9Faag&F37$Ffr$\"3bM/wH\"3C'pF37$F[s$\"3)3nOM!3%zL)F37$F `s$\"3<<;C9=bg**F37$Fes$\"3!*p&*[1Q')z6F07$Fjs$\"3jGqVpN`_8F07$F_t$\"3 7,<&pShAb\"F07$Fdt$\"3A>/'34RIx\"F07$Fit$\"3*)\\.aUsA.?F07$F^u$\"3]c+[ aWJRAF07$Fcu$\"3*\\Te3%G&o^#F07$Fhu$\"3gu:e,#f+y#F07$F]v$\"3-Ag^kV`vIF 07$Fbv$\"3'RgRzYjhN$F07$Fgv$\"3oSK1ZD)pn$F07$F\\w$\"3$Qe$y6#=A*RF07$Fa w$\"3AMgHnreNVF07$Ffw$\"3VzHC_h)\\o%F07$F[x$\"31%GUz`!Gl]F07$F`x$\"3Y2 ta&)odXaF07$Fex$\"3?lEe&**>([eF07$Fjx$\"3amy*f6TFE'F07$F_y$\"3K^[pE\\m bmF07$Fdy$\"3C7Y*yL,27(F07$Fiy$\"3d5US2$Q*\\vF07$F^z$\"3-\"*)R!))3V@!) F07$Fcz$\"3$f)3\"*pS3'[)F07$Fhz$\"\"*F*-F][l6&F_[lFb[lF`[lFb[l-F$6$7S7 $F($!\"%F*7$F.$!3A!e4#)QZ)eOF07$F5$!3&*[e7a6\"F37$Far$ !3CH$*>9#z`J'Ffn7$Ffr$!38sw$4j*)>u#Ffn7$F[s$!37a/+x'ova(Fao7$F`s$!3fBH SG34)*Q!#B7$Fes$!3[xsBc\")3LuFao7$Fjs$!3'*)*GqVMScEFfn7$F_t$!3SBo$=Oul /'Ffn7$Fdt$!3,&>/'3\")G*4\"F37$Fit$!3q1NSD.>D#pF37$Fgv$!3:3Cjqg!*=%)F37$F\\w$!3))y\"p6+56'**F37$Faw$!3EMgH- E6\"F37$Far$ \"3CH$*>9#z`J'Ffn7$Ffr$\"38sw$4j*)>u#Ffn7$F[s$\"37a/+x'ova(Fao7$F`s$\" 3fBHSG34)*QFiil7$Fes$\"3[xsBc\")3LuFao7$Fjs$\"3'*)*GqVMScEFfn7$F_t$\"3 SBo$=Oul/'Ffn7$Fdt$\"3,&>/'3\")G*4\"F37$Fit$\"3q1NSD.>D#pF37$Fgv$\"3:3Cjqg!*=%)F37$F\\w$\"3))y\"p6+56' **F37$Faw$\"3EMgH-EeqT')f\">DF07 $Fbq$!3oINw)[n]C#F07$Fgq$!3gMKdOph:?F07$F\\r$!3$\\9d@wl:F07$Ffr$!3/5Fkz*)fe8F07$F[s$!3sL.,d5I\"=\"F07$F`s$!34&\\dZh_R+ \"F07$Fes$!3bY!H#)pB+N)F37$Fjs$!3]%HqV*\\%f+(F37$F_t$!3&HN]G!3q'o&F37$ Fdt$!3$y>/'3`=oWF37$Fit$!3+:NSD#3\"=MF37$F^u$!3%)z0![/)yNDF37$Fcu$!3;l Te3)f,r\"F37$Fhu$!3lBC[#)Qb16F37$F]v$!3h`Ag^kPlgFfn7$Fbv$!3P*z$HF^&H#G Ffn7$Fgv$!3?64Cjqm)z'Fao7$F\\w$!34kc._-X)y$Fiil7$Faw$!3(ewMgHP!enFao7$ Ffw$!3\"))ok4*=Y0FFfn7$F[x$!331&GUzG4G'Ffn7$F`x$!3V-k!))QHF6\"F37$Fex$ !3-*)*f\"*)=p]eqT')f\">DF07$Fbq$\"3oINw)[n]C#F 07$Fgq$\"3gMKdOph:?F07$F\\r$\"3$\\9d@wl:F07$Ffr $\"3/5Fkz*)fe8F07$F[s$\"3sL.,d5I\"=\"F07$F`s$\"34&\\dZh_R+\"F07$Fes$\" 3bY!H#)pB+N)F37$Fjs$\"3]%HqV*\\%f+(F37$F_t$\"3&HN]G!3q'o&F37$Fdt$\"3$y >/'3`=oWF37$Fit$\"3+:NSD#3\"=MF37$F^u$\"3%)z0![/)yNDF37$Fcu$\"3;lTe3)f ,r\"F37$Fhu$\"3lBC[#)Qb16F37$F]v$\"3h`Ag^kPlgFfn7$Fbv$\"3P*z$HF^&H#GFf n7$Fgv$\"3?64Cjqm)z'Fao7$F\\w$\"34kc._-X)y$Fiil7$Faw$\"3(ewMgHP!enFao7 $Ffw$\"3\"))ok4*=Y0FFfn7$F[x$\"331&GUzG4G'Ffn7$F`x$\"3V-k!))QHF6\"F37$ Fex$\"3-*)*f\"*)=p] " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 22 "Preparation po lynomial" }}{PARA 0 "" 0 "" {TEXT -1 291 "Some sophisticated results o f differential algebra allow us to say that the properties of singular solution can be determined in the way the equation of the singular so lution appear algebraically in the differential equation under conside ration. This algebraic structure is revealed by the p" }{TEXT 273 21 " reparation polynomial" }{TEXT -1 42 " that can be computed by a comman d of the " }{TEXT 272 7 "diffalg" }{TEXT -1 9 " package." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "We illustrate here the fact that the preparation polynomial is not unique." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "R1 := differential_ring(ranking=[y, z], derivations=[s,t]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "R2 := differential_ring(ranking=[y, z], derivations=[t,s]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "p := y[s,t]*y[s,s]+y[t,t]^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"p G,&*&&%\"yG6$%\"sG%\"tG\"\"\"&F(6$F*F*F,F,*$)&F(6$F+F+\"\"#F,F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "a := y[s]+y[t];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG,&&%\"yG6#%\"sG\"\"\"&F'6#%\"tGF*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "preparation_polynomial(p,z[] =a, R1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&&%\"zG6#%\"sG\"\"\"&F& 6#%\"tGF)F)*$)F*\"\"#F)!\"\"*&&%\"yG6$F,F,F)F%F)F0*(F/F)F2F)F*F)F)" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "preparation_polynomial(p,z[ ]=a, R2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*$)&%\"zG6#%\"sG\"\"#\" \"\"F+*(F*F+F&F+&F'6#%\"tGF+!\"\"*$)F-F*F+F+*&&%\"yG6$F)F)F+F&F+F0*(F* F+F4F+F-F+F+" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 33 "Computing a min imal decomposition" }}{PARA 0 "" 0 "" {TEXT -1 320 "From a decompositi on of the solution set by diffalg[Roenfeld_Groebner] we can determine \+ a minimal decomposition with the following ingredients: the computatio n of preparation polynomials, application of the regular component the orem (E.Hubert 99), the low power theorem for regular differential pol ynomials (Hubert 99). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "We illustrate the process in two examples." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 55 "A n ordinary differential equation (back to tobogganing)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(di ffalg);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7=%3Rosenfeld_GroebnerG%+be longs_toG%-delta_leaderG%1delta_polynomialG%'denoteG%,derivativesG%2di fferential_ringG%3differential_spremG%.differentiateG%*equationsG%5ess ential_componentsG%0field_extensionG%(greaterG%,inequationsG%(initialG %3initial_conditionsG%.is_orthonomicG%'leaderG%6power_series_solutionG %7preparation_polynomialG%.print_rankingG%%rankG%(reducedG%-reduced_fo rmG%.rewrite_rulesG%)separantG%(versionG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "R := differential_ring(ranking=[y,z],derivations=[t]) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "p := y[t]^2+2/3*y[]^3- 2*y[]^2+3/2*y[]-1/3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG,,*$)&% \"yG6#%\"tG\"\"#\"\"\"F-*&#F,\"\"$F-)&F)6\"F0F-F-*&F,F-)F2F,F-!\"\"*&# F0F,F-F2F-F-#F-F0F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "G := equations(Rosenfeld_Groebner([p],R));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG7$7#,,*$)&%\"yG6#%\"tG\"\"#\"\"\"\"\"'*&\"\"%F/)&F+6\"\"\" $F/F/*&\"#7F/)F4F.F/!\"\"*&\"\"*F/F4F/F/F.F:7#,(*$F9F/F.*&\"\"&F/F4F/F :F.F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "a := 2*y[]^2-5*y[] +2; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG,(*$)&%\"yG6\"\"\"#\"\" \"F+*&\"\"&F,F(F,!\"\"F+F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "pp := preparation_polynomial(p, z[]=a,R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ppG,(*&,&&%\"yG6\"\"#;\"\")!\"\"\"\"\")&%\"zGF*\"\"# F.F.*&\"\"'F.)&F16#%\"tGF2F.F.*&,&!\"*F.*&\"#=F.F(F.F.F.F0F.F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "hat_a := gcd( coeff(pp, z[]) , a); tilde_a := normal(a / hat_a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%&hat_aG,&!\"\"\"\"\"*&\"\"#F'&%\"yG6\"F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(tilde_aG,&&%\"yG6\"\"\"\"\"\"#!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " By LPT" } {XPPEDIT 18 0 "\{tilde_a\} = \{y-2\};" "6#/<#%(tilde_aG<#,&%\"yG\"\"\" \"\"#!\"\"" }{TEXT -1 37 " is an essential regular component. " }} {PARA 0 "" 0 "" {TEXT -1 12 "What about " }{XPPEDIT 18 0 "\{hat_a\} = \{2*y-1\};" "6#/<#%&hat_aG<#,&*&\"\"#\"\"\"%\"yGF*F*F*!\"\"" }{TEXT -1 4 " ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 45 "pp := preparation_polynomial(p, z[]=hat_a,R);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ppG,(*$)&%\"zG6\"\"\"$\"\"\"F,*&F+F ,)F(\"\"#F,!\"\"*&F+F,)&F)6#%\"tGF/F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "By LPT (Hubert 97) " } {XPPEDIT 18 0 "\{2*y-1\};" "6#<#,&*&\"\"#\"\"\"%\"yGF'F'F'!\"\"" } {TEXT -1 15 " is redundant. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 152 "The essentiality or redundancy properties of t he singular solutions reflect in the behaviour of the non-singular sol utions around the singular solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "y-2 = 0;" "6#/,&%\"yG\"\"\"\"\"#!\"\" \"\"!" }{TEXT -1 46 " is an envelope of the non-singular solutions." } }{PARA 0 "" 0 "" {XPPEDIT 18 0 "2*y-1 = 0;" "6#/,&*&\"\"#\"\"\"%\"yGF' F'F'!\"\"\"\"!" }{TEXT -1 38 " is a limit of non-singular solutions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "plot(\{2,1/2,seq(2-3/2*tanh(1/2*(y-C))^2,C=-4..3), seq(2-3/2* coth(1/2*(y-C))^2,C=-4..3) \},y=-4..4, -3/2..5/2);" }}{PARA 13 "" 1 " " {GLPLOT2D 332 273 273 {PLOTDATA 2 "66-%'CURVESG6$7S7$$!\"%\"\"!$\"\" #F*7$$!3nmmmmFiDQ!#+++:v2*\\#F0F+7$$!3BLLL8>1DBF0F+7$$!3lmmmw))yr@F 0F+7$$!3:+++S(R#**>F0F+7$$!30++++@)f#=F0F+7$$!3-+++gi,f;F0F+7$$!3qmmm \"G&R2:F0F+7$$!3WLLLtK5F8F0F+7$$!3fLLL$HsV<\"F0F+7$$!3+-++]&)4n**!#=F+ 7$$!37PLLL\\[%R)F^oF+7$$!3F)*****\\&y!pmF^oF+7$$!3Y******\\O3E]F^oF+7$ $!3NKLLL3z6LF^oF+7$$!3sLLL$)[`P(******z -6j'F^oF+7$$\"3q\"******4#32$)F^oF+7$$\"3r$*****\\#y'G**F^oF+7$$\"3H** ****H%=H<\"F0F+7$$\"35mmm1>qM8F0F+7$$\"3%)*******HSu]\"F0F+7$$\"3(HLL$ ep'Rm\"F0F+7$$\"3()******R>4N=F0F+7$$\"3$emm;@2h*>F0F+7$$\"3^*****\\c9 W;#F0F+7$$\"3Lmmmmd'*GBF0F+7$$\"3j*****\\iN7]#F0F+7$$\"3aLLLt>:nEF0F+7 $$\"36LLL.a#o$GF0F+7$$\"3ammm^Q40IF0F+7$$\"3z******z]rfJF0F+7$$\"3gmmm c%GpL$F0F+7$$\"3/LLL8-V&\\$F0F+7$$\"3>+++XhUkOF0F+7$$\"3=+++:oFfu7$FAFfu7$FDFfu7$FGFfu7$F JFfu7$FMFfu7$FPFfu7$FSFfu7$FVFfu7$FYFfu7$FfnFfu7$FinFfu7$F\\oFfu7$F`oF fu7$FcoFfu7$FfoFfu7$FioFfu7$F\\pFfu7$F_pFfu7$FcpFfu7$FfpFfu7$FipFfu7$F \\qFfu7$F_qFfu7$FbqFfu7$FeqFfu7$FhqFfu7$F[rFfu7$F^rFfu7$FarFfu7$FdrFfu 7$FgrFfu7$FjrFfu7$F]sFfu7$F`sFfu7$FcsFfu7$FfsFfu7$FisFfu7$F\\tFfu7$F_t Ffu7$FbtFfu7$FetFfu7$FhtFfu-F[u6&F]uFauF^uFau-F$6$7Y7$F($\"3K2uC+M)))R &F^o7$F.$\"3O7z%Q)QmtaF^o7$F2$\"3+7.q*>M)\\bF^o7$F5$\"34=[\"\\M$*)\\cF ^o7$F8$\"3zEug&4A&odF^o7$F;$\"3u'e%z8%)R2fF^o7$F>$\"3]P*)fufkdgF^o7$FA $\"3I&)RpBQKQiF^o7$FD$\"3pL]G/')*fX'F^o7$FG$\"3d\"\\w)fXm3nF^o7$FJ$\"3 '=i1p\"p%4,(F^o7$FM$\"3;M=yTXJ S%o\")F^o7$FV$\"386Tk++Ap')F^o7$FY$\"3U]uD5F!*z\"*F^o7$Ffn$\"3HW#*HAi& *f)*F^o7$Fin$\"3xnu-up$*\\5F07$F\\o$\"35$oIFIT:8\"F07$F`o$\"3dZl4HN*)4 7F07$Fco$\"3+9i]Vjk,8F07$Ffo$\"3v'*z#z['R$R\"F07$Fio$\"3g3]\")p'*y\"\\ \"F07$F\\p$\"3)>u%zh^Q#e\"F07$F_p$\"3;AqgDt^x;F07$Fcp$\"3(R#[`p%y)ph**zvuT=F07$Fip$\"31*HH;epr!>F07$F\\q$\"3RlcV.=Be>F07$$\"3W %*****\\C4puF^o$\"3]!)\\*H`Li(>F07$F_q$\"3Z&3?st.$*)>F07$$\"3q#****\\< !)y6*F^o$\"3JX!yCx&3(*>F07$Fbq$\"3-M;pC4)***>F07$$\"3N***\\iZ!)y.\"F0$ \"3-!fK(H?Y**>F07$$\"3L****\\FJ*G3\"F0$\"3f>vb@iU(*>F07$$\"3J***\\(yd! z7\"F0$\"3A7a4J<)Q*>F07$Feq$\"3n6b)*[G%)))>F07$$\"3qKLLo,\"QD\"F0$\"3@ 8,#*p'*4w>F07$Fhq$\"3o7P-uDwe>F07$F[r$\"3B^];voV2>F07$F^r$\"3%pk:a/5h% =F07$Far$\"3>u1$\\\"F07$F]s$\"31J-Eyk<%R\"F07$F`s$\"3$=vR$>G^,8F07$F cs$\"3dt!z\")pX7@\"F07$Ffs$\"3Wr$Gnj?v7\"F07$Fis$\"3a&)[x5&yj0\"F07$F \\t$\"3$)4\"*yYq\"3#)*F^o7$F_t$\"3Hzj'ow`DA*F^o7$Fbt$\"3DI8k2<+_')F^o7 $Fet$\"3nw<$zI#*y;)F^o7$Fht$\"3@HZ&Qe*f5xF^o-F[u6&F]uF^uF^uFau-F$6$7en 7$F($\"3)eY(zsBwfgF^o7$F.$\"3ikP([?]ID'F^o7$F2$\"3PgkVa`?[kF^o7$F5$\"3 P!R.aMP>q'F^o7$F8$\"3'e&*yOe?*)*pF^o7$F;$\"3X&z?1b]8M(F^o7$F>$\"37TP&y N;bq(F^o7$FA$\"3j5+**4e'[8)F^o7$FD$\"3Z$pA!*[p)R')F^o7$FG$\"3kMu)3_:&4 #*F^o7$FJ$\"3Sxe1[$=\"o)*F^o7$FM$\"3REmEv%o50\"F07$FP$\"3Rck_LjKI6F07$ FS$\"3#H;'yJ-&o@\"F07$FV$\"3aY(yl%y'fI\"F07$FY$\"3G&RI`(zn!R\"F07$Ffn$ \"3f>xQkE9%\\\"F07$Fin$\"3&o&pWN<.#e\"F07$F\\o$\"3yA*H)eSY\"o\"F07$F`o $\"3;&p2r#f%Rw\"F07$Fco$\"3StK/K+%[%=F07$Ffo$\"3T*p&eL&>\"4>F07$Fio$\" 3:F07$$!3/LLLeGmCDF^o$\"3KuWdM%\\j(>F07$F\\p$\"3\"ex\"4q`t))> F07$$!39nm;aH-88F^o$\"34Y\"o;U`N*>F07$$!3l0++]-6&)))!#>$\"3??hXYM/(*>F 07$$!3&)RLLe4**RYFhil$\"3J-#eF$H>**>F07$F_p$\"3W_*yG:%****>F07$$\"3S6L L3Uh9SFhil$\"3D#\\U+x&R**>F07$$\"3&oHLL3+TU)Fhil$\"3cKgcR>M(*>F07$$\"3 BGL$efeLG\"F^o$\"3vXLqH1%Q*>F07$Fcp$\"3Ll@u'Q0*))>F07$$\"3Lhmmm7+#\\#F ^o$\"3@J`$R;^p(>F07$Ffp$\"3]ZbG>\"\\3'>F07$Fip$\"3M5$)z\\g$G\">F07$F\\ q$\"3S^[Cl!R\"F07$F^r$\"3 qz9:acD.8F07$Far$\"3:pZ\">0S@@\"F07$Fdr$\"33&f))*R4$=8\"F07$Fgr$\"3XbLN *HhkF^o7$Fet$\"3&H#=n_dQ_iF^o7$FhtFddl-F[u6&F]uFauFauF^u-F$6$7W7$F($\" 3:R5U7:'*H6F07$F.$\"33oZ.pj.<7F07$F2$\"3#pM[)4n#yH\"F07$F5$\"3Y@l@u=-$ R\"F07$F8$\"3z5is\"p+;\\\"F07$F;$\"3+QOYH8\"**e\"F07$F>$\"3M%))H+vS&y; F07$FA$\"3W7/x>wokF07$FJ$\" 3Kl,x\\I1h>F07$$!3%******\\RD%[AF0$\"3/))pr\"z#4x>F07$FM$\"3\"frKUU()* ))>F07$$!3SLLL3V^&3#F0$\"3aXKTx5E(*>F07$FP$\"3]&z^Ky*****>F07$$!35+++? 4h7>F0$\"3UN_\">#)Rr*>F07$FS$\"3P\\=OK7q))>F07$$!3/+++!=*\\UF07$FV$\"35%\\+B**Hs&>F07$FY$\"3PBL*QJgD\">F07$Ffn$\"341VC$QTA%= F07$Fin$\"3(RO<%QN%3x\"F07$F\\o$\"3GVplXo(yn\"F07$F`o$\"3oQ*G]*p\"**e \"F07$Fco$\"3OIay*G&o!\\\"F07$Ffo$\"3[<.$pdhjR\"F07$Fio$\"3TC64?-g+8F0 7$F\\p$\"3W..Q\"G8n@\"F07$F_p$\"3m.y1Cy&=8\"F07$Fcp$\"3==$*>.sy]5F07$F fp$\"3`qKl\\!)[k)*F^o7$Fip$\"37&**[:rybB*F^o7$F\\q$\"3a&>&zAN=c')F^o7$ F_q$\"3\"fl$3r^;b\")F^o7$Fbq$\"3bD(p$>Y'F^o7$ Far$\"3^!*>\")o-uTiF^o7$Fdr$\"3FKBN)eYP1'F^o7$Fgr$\"3j7#=3i:S!fF^o7$Fj r$\"3\")Qz'4FE/x&F^o7$F]s$\"3!4;l`N'=^cF^o7$F`s$\"3Q*)>xkm[`bF^o7$Fcs$ \"3'RBhqEq%oaF^o7$Ffs$\"3fa'Rdq$)oR&F^o7$Fis$\"3hyZS$ov1M&F^o7$F\\t$\" 3!=!\\l..)eG&F^o7$F_t$\"35uN;t\">VC&F^o7$Fbt$\"3Wd#=K:$f1_F^o7$Fet$\"3 cB_q]1#f<&F^o7$Fht$\"3<.;[d0*z9&F^o-F[u6&F]uF^uFauF^u-F$6$7W7$F($\"36 \"*)[%*fr'z;F07$F.$\"3jP[@#*f%3x\"F07$F2$\"3hu2>w_!=%=F07$F5$\"3o@Qc$p !*)3>F07$F8$\"31NM*z/K&f>F07$$!3******\\UR.YKF0$\"3&H![k,t_x>F07$F;$\" 3'Qh^Z*QP!*>F07$$!3N+++0vJ\"3$F0$\"3*>Qi!HI_(*>F07$F>$\"3'4THwQ)****>F 07$$!34LLL[5-?HF0$\"3_j_sF07$FA$\"327<7(f(>!*>F07$$!3dLLL3(4Jv#F 0$\"39#H#fs@Px>F07$FD$\"3V?I+]QZf>F07$FG$\"3\"\\#)>cX.(4>F07$FJ$\"3lWe sP=NT=F07$FM$\"3([*3ua\"f&p-Ws7jkF^o7$F_q$ \"3m+^t1S'pC'F^o7$Fbq$\"3%\\!**QNI2ngF^o7$Feq$\"3j!z`.\"Hf'*eF^o7$Fhq$ \"3,L&*4jO8mdF^o7$F[r$\"3^J%HpKYsk&F^o7$F^r$\"3_O\\&HN>_b&F^o7$Far$\"3 \"peGEBq#paF^o7$Fdr$\"3T`]I*G=/S&F^o7$Fgr$\"3\"HkUmf'4R`F^o7$Fjr$\"3( \\&Rh4P9)G&F^o7$F]s$\"3&)z!GuS;HC&F^o7$F`s$\"3x.K>eY.1_F^o7$Fcs$\"31sk KE\"oS<&F^o7$Ffs$\"3ZM%*)=LUs9&F^o7$Fis$\"3Y1)4]IPi7&F^o7$F\\t$\"32]>2 W*3e5&F^o7$F_t$\"3)H!\\m2kM!4&F^o7$Fbt$\"3xLX5]WLw]F^o7$Fet$\"3RY7'Q') e\\1&F^o7$Fht$\"3/6t!3F8Y0&F^o-F[u6&F]uFauF^uF^u-F$6$7UF'7$$!3cLLL$Q6G \"RF0$\"3T*)zH7H:(*>F07$F.$\"3U&=Izva'))>F07$$!36++]2y/ ww>F07$F2$\"3cs4Wmy\"3'>F07$F5$\"3(3un#zw96>F07$F8$\"3u^\"f_e`T%=F07$F ;$\"3)[0#*=\"4&Rw\"F07$F>$\"3))R*Rv`,3o\"F07$FA$\"3Ttl*))[t!*e\"F07$FD $\"3PT?R%fg9\\\"F07$FG$\"3gzZIHWN%R\"F07$FJ$\"3n>pt(3fsH\"F07$FM$\"3(H 5Aw;'p:7F07$FP$\"3+(3_g#ofH6F07$FS$\"32o%Gd4#4]5F07$FV$\"3T^yytGt/)*F^ o7$FY$\"3H([kTbwEB*F^o7$Ffn$\"3p(>V#G#y^i)F^o7$Fin$\"3++MF#eP%p\")F^o7 $F\\o$\"3FR/,>z`-xF^o7$F`o$\"3!fdeR$>PTtF^o7$Fco$\"3.WNSO;(f*pF^o7$Ffo $\"314fI_\"y8r'F^o7$Fio$\"3JBFSX,@bkF^o7$F\\p$\"323c`\">-BD'F^o7$F_p$ \"3^O%)3NR!Q1'F^o7$Fcp$\"3E\"*p?;q,(*eF^o7$Ffp$\"3nm\")oyxnsdF^o7$Fip$ \"3i#yQ-#eAdcF^o7$F\\q$\"3RI'obbF^o7$F_q$\"3n))=_b#*HraF^o7$Fbq$\" 3-W'Qy)**p,aF^o7$Feq$\"3J[0))evDO`F^o7$Fhq$\"3([HXWU6lG&F^o7$F[r$\"3^( *=M(*fUT_F^o7$F^r$\"3.8u#QQ(o1_F^o7$Far$\"33*er#o$oV<&F^o7$Fdr$\"3&po# e5\\c[^F^o7$Fgr$\"3'*yzdIzkD^F^o7$Fjr$\"3V5(o'==l1^F^o7$F]s$\"3uh1M(* \\#)*3&F^o7$F`s$\"3x3^y'=Fh2&F^o7$Fcs$\"3eS=A*GsU1&F^o7$Ffs$\"3Qb#z#*G OV0&F^o7$Fis$\"3?3UZWTcY]F^o7$F\\t$\"3)4w#\\;?,R]F^o7$F_t$\"3l)[IM/+L. &F^o7$Fbt$\"3*p*pcar7G]F^o7$Fet$\"3^s]B3)HR-&F^o7$Fht$\"3V(QX-E9,-&F^o Fjt-F$6$7go7$F($\"3)4[)=F'>-f%F^o7$F.$\"308B8R7*3^%F^o7$F2$\"3:m:(QSW# HWF^o7$F5$\"30(>qk-u1K%F^o7$F8$\"3Fw7g2l(**=%F^o7$F;$\"3#*p$)ogd$\"3ss/$\\fA@'QF^o7$FA$\"3dMPKhuC]OF^o7$FD$\"3msx*=czuQ$F^o7$FG$\" 3*3(o)>Yy;2$F^o7$FJ$\"3\"HG@<6@xn#F^o7$FM$\"34^cvdxFfAF^o7$FP$\"3`28;E &f()o\"F^o7$FS$\"3dl)zYC`1$)*Fhil7$FV$\"3A+,CO1&eU\"Fhil7$FY$!3GL'p)4U PYzFhil7$Ffn$!3j7F:9LD*=#F^o7$Fin$!3SX!>fMREo$F^o7$F\\o$!3/?h*3%y'z!fF ^o7$F`o$!3]Jta&purZ)F^o7$Fco$!3s)p!pwK'=A\"F07$Ffo$!3jhyyPt<4r(\\F07$Fcp$!3#pFb.O Uux(F07$Ffp$!3,hkh0#z**\\(=;y4&)F ^o$!3_kyb![%z\"p#Feeo7$$\"3?#***\\P6[7()F^o$!3W!fw!yjY4OFeeo7$$\"3X#** \\il!=:*)F^o$!3'HMxBLK%)3&Feeo7$F]_l$!3-zWu-Cu+xFeeo7$$\"3'H**\\Ppz0K* F^o$!3hk#p'[Jz)H\"!#97$$\"3@$***\\7#zK_*F^o$!3:_3ZJk6REF]go7$$\"3L$*\\ (=(*GYi*F^o$!32!*)eXEOsD%F]go7$$\"3Y$**\\7tyfs*F^o$!3y8ZW*)4m*)zF]go7$ $\"3_$\\P4h`mx*F^o$!3wMIo!*)*p-7!#87$$\"3e$*\\i!\\Gt#)*F^o$!3K\"oZ%y%z A,#Fbho7$$\"3l$\\7.P.!y)*F^o$!3Hkyf^rHJSFbho7$Fbq$!3#*f*=AK:&z6!#77$$ \"38$\\iS1W\\)**F^o$!314U-I!))ok#!#67$$\"3D*\\7y)47/5F0$!3k(e]/HII`$F` io7$$\"3U\\(=#pvu45F0$!3TCUAy`o9jFbho7$$\"3O**\\i]TP:5F0$!3]DU.%=_$QDF bho7$$\"3I\\7.K2+@5F0$!3KD;Lq$\\.O\"Fbho7$$\"3Z*\\PMJFm-\"F0$!3#4bhzYe 9Y)F]go7$$\"3T\\P%[*QDK5F0$!3*Qw\"fR4\\mdF]go7$Fe_l$!3D&GFeeo7$Fg`l$!3H!oIX#f49#*Fhdo7$Fhq$ !3a*)yMU'3iD&Fhdo7$F[r$!3dc0.)ft2B#Fhdo7$F^r$!3sbVuPP3i7Fhdo7$Far$!3W/ \\hW@h?wF07$Fdr$!3dNGfV&\\32&F07$Fgr$!37&*R.ArQdMF07$Fjr$!3#Q.[V.g%QCF 07$F]s$!3dx])z&G&Rr\"F07$F`s$!3Oat/ztC@7F07$Fcs$!3!=N,**Q$*f_)F^o7$Ffs $!3%e%y-ybd)y&F^o7$Fis$!3?9!)Q!\\1V%QF^o7$F\\t$!3R;*3?;NR5#F^o7$F_t$!3 rr-:L(3$p()Fhil7$Fbt$\"3?za**en9F?F]go7$$!3'******p#y0SRF0$!3&z eez,%))o;F]go7$$!3*******z`3Y$RF0$!3v7EwRk;-9F]go7$$!3-+++\\#f\"HRF0$! 3fORGduf%>\"F]go7$$!38+++h*4P#RF0$!3dkIUfy*)H5F]go7$$!3s*****>ng#=RF0$ !3W:=X!4m-(*)Feeo7$$!3%)*****RQ6G\"RF0$!3?Y'QB_7G)yFeeo7$$!3&******f4i t!RF0$!3;RAX]wb\")pFeeo7$$!3)******p!G\">!RF0$!3ww[:NY'*QF0$!33+W*\\^?re&Feeo7$$!38+++IU,\"*QF0$!3ejAWmu%QF0$!3=t&y'*4]sc#Feeo 7$$!3')*****Hjq>%QF0$!3P\"='R@Sd#R#Feeo7$$!3(******\\M@l$QF0$!3:Mg%o!G 2NAFeeo7$$!3++++c?2JQF0$!3?JE*pWmD4#Feeo7$$!3/+++nFiDQF0$!3!yZscA5K'>F eeo7$$!3.+++r8)3#QF0$!3B@A&>9H,'=Feeo7$$!3$******R(*Rh\"QF0$!3X[2,$e> \\w\"Feeo7$$!3%)*****pd)R6QF0$!3*)RO\\a+!on\"Feeo7$$!3?+++!=dm!QF0$!3k r?!en$3&f\"Feeo7$$!3,+++'Qurz$F0$!3V'poAk2&[9Feeo7$$!3\"******Hf\"p(y$ F0$!34\"4w4?I6K\"Feeo7$$!34+++)z3#yPF0$!3a4l3G#R(47Feeo7$$!3*******\\+ E(oPF0$!36hT(>Rn<6\"Feeo7$$!3D+++6KCfPF0$!3y=K%yrU^-\"Feeo7$$!3:+++=/w \\PF0$!3G;)3)zEy\"[*Fhdo7$$!3()*****4$[zIPF0$!3sgjorgIz\")Fhdo7$$!3%** ****HCH=r$F0$!3t4N(o\"HVDrFhdo7$$!36+++cO'Gp$F0$!37rrg.%*pgiFhdo7$$!3u *****z1)*Qn$F0$!3+m.8kJTUbFhdo7$$!37+++&\\S7j$F0$!3$Ra2:!ej7VFhdo7$$!3 1+++AHe)e$F0$!3\\N&*\\l&y^W$Fhdo7$$!3++++\\`#fa$F0$!3\\9&Q4PO0\"GFhdo7 $$!3.+++xxE.NF0$!3k_OBn%R.YKF0$!3cR^S=-ko&*F07$$!3; +++[9cgJF0$!3;)4=bl:>`(F07$$!33+++iN2-IF0$!3^*f:!Qh!*[]F07$FA$!32*zV)* eJaZ$F07$$!32+++\")3DoEF0$!3u,0F\"*eVCCF07$FG$!3H$QEK4W]r\"F07$$!3()** ***H\">1DBF0$!34]O`!)*[#Fhil7$$!3,++ +$HsV<\"F0$\"3*HI!Gy2i9)*Fhil7$$!31+++S&)4n**F^o$\"394,Mz=a.cDAF^o7$$!3n******R&y!pmF^o$\"3m,\"[$R#pwp#F^o7 $$!3u******RO3E]F^o$\"3yAljH;AoIF^o7$$!3&)******H3z6LF^o$\"3/Vqr(fY%)Q $F^o7$$!36+++!)[`P2Vs\"F^o$\"3'[U;[k#eEJ%F^o7$$\"3C+++5G5JmF^o$ \"3'QdVI#f$HU%F^o7$$\"3I+++5@32$)F^o$\"3Zlkzz@T8XF^o7$$\"34+++g#y'G**F ^o$\"3!=^iMZYse%F^o7$$\"3/+++J%=H<\"F0$\"3j[\"[TkJgl%F^o7$$\"3*******z !>qM8F0$\"3KDPYB%4zq%F^o7$$\"3$******4ISu]\"F0$\"3m44mqYiaZF^o7$$\"38+ ++fp'Rm\"F0$\"3C*prp%[U!z%F^o7$$\"3&******4%>4N=F0$\"3iuI8P3eB[F^o7$$ \"3;+++8s5'*>F0$\"3\\YNuM*[*\\[F^o7$$\"3/+++mXTk@F0$\"3i3nL&o!Ht[F^o7$ $\"3x*****zwl*GBF0$\"3t!>YQW%e#*[F^o7$$\"3;+++EcB,DF0$\"3vu#)>gQj4\\F^ o7$$\"3!******\\(>:nEF0$\"3oPM&f[%[B\\F^o7$$\"3\"******\\SDo$GF0$\"3L2 bkI6XN\\F^o7$$\"3)******H&Q40IF0$\"3'fs#QmhYX\\F^o7$$\"3()*****43:(fJF 0$\"3N>%)[d3H`\\F^o7$$\"3/+++e%GpL$F0$\"3%H?#\\ci)3'\\F^o7$$\"3%)***** \\@Ia\\$F0$\"3`b3&f'eim\\F^o7$$\"3\"******p9EWm$F0$\"3D,!\\R+?=(\\F^o7 $$\"3!******p\"o.w\\F^o7$Fht$\"3&*)f18te)z\\F^oF^dl-F$ 6$7co7$F($\"3.$yKM\"[`][F^o7$F.$\"3pjJCBC*=#[F^o7$F2$\"3>MtMM,^#z%F^o7 $F5$\"3&>xGWI!f`ZF^o7$F8$\"33g<4wa'pq%F^o7$F;$\"3jue/%\\0$\" 3jp-)Q%*z5f%F^o7$FA$\"34t\\)RR))p^%F^o7$FD$\"3z%3e=&3&fU%F^o7$FG$\"3Jf N&*\\tvN@-!* GxeQF^o7$FS$\"3`p!RE%fCLOF^o7$FV$\"3q-(>:VGD-u#)Fhil 7$Fio$!3ZI?()*)or`@F^o7$F\\p$!347%\\*fNR*o$F^o7$F_p$!3%Ho3$=$*y/eF^o7$ Fcp$!3soWF#G=+q)F^o7$Ffp$!3W:?:Wu+/7F07$Fip$!3Rf-Aiw%)*o\"F07$F\\q$!3P i!QGwn()R#F07$F_q$!3l&*\\`$)Q!3U$F07$Fbq$!3M\"*oo!=$pR\\F07$Feq$!3Y?d: f$exy(F07$Fhq$!3)og)H^skc7Fhdo7$F[r$!3`)H36*>TY*Fhdo7$Far$!32l'eji@j>#Feeo7$$ \"3Omm\"zvX`(=F0$!3e#[>crN8&QFeeo7$$\"3&GLLed*f:>F0$!3()Ho$f^3HT)Feeo7 $$\"3?m;z%[Ed$>F0$!3(pke(GJS^9F]go7$$\"3M***\\PR`e&>F0$!3kLoU?PixIF]go 7$$\"3,m\"H#[o\"f'>F0$!3L&[7u+F0$!360fSO*e)R5 Fbho7$$\"3#f\"z%*H?,\")>F0$!3tg5$Q`cSm\"Fbho7$$\"3:*\\(=dP/')>F0$!3qB; ov.N!3$Fbho7$$\"3Q#3FW[v5*>F0$!3\"y)*4W[UJ`(Fbho7$Fdr$!3tZWjeERfRF`io7 $$\"3[K$ek\"oO,?F0$!3HaD6F&o;@$Ffio7$$\"38***\\7UEm+#F0$!3-a$o`=QkO\"F `io7$$\"3yl;/Eg)=,#F0$!3r9y75t&oC%Fbho7$$\"3VKL$3jXr,#F0$!3[w-0]h!4/#F bho7$$\"3tlmTS[mF?F0$!3EdC\"\\?='QyF]go7$$\"3.******\\S=Q?F0$!3(>P)[)> lT6%F]go7$$\"3LKLefKq[?F0$!3X7h&)GI]GDF]go7$$\"3jlm;pCAf?F0$!3(zq!3`'4'e4AFeeo7$ $\"3#HLLe;!pYAF0$!3!zj-,QS\"G7F07$Fet$!3;dXo%[Pks)F^o7$Fht$!3\"Hm%\\9\\#4'eF^oF]`m -F$6$7Y7$F(F\\dl7$F.$\"3\"[$p9([^%p\")F^o7$F2$\"3Zx21'G=?i)F^o7$F5$\"3 \\U:cWjd%>*F^o7$F8$\"3=yo\"*ehQU)*F^o7$F;$\"3$Gt%p0^+c5F07$F>$\"3vB3pq r'*G6F07$FA$\"3^7Peail57F07$FD$\"3$)z)*[s=\"4I\"F07$FG$\"3k39fvIS&R\"F 07$FJ$\"3*)*3R]>@`\\\"F07$FM$\"3o&Q\"f4q]$e\"F07$FP$\"3PM!37*f3!o\"F07 $FS$\"3hf??#[n1x\"F07$FV$\"3m)Rt@()Q#[=F07$FY$\"3)yB7Th_u!>F07$Ffn$\"3 i_\"zI6\"eg>F07$$!3_LLL$yP2D\"F0$\"3Z\")R`9)om(>F07$Fin$\"3(Gj#[(Rb')) >F07$$!38++v$oc*H6F0$\"3sHy\"p]%o$*>F07$$!3!pmmT2Tb3\"F0$\"3U-#yYOfs*> F07$$!3mLLeka7T5F0$\"3w;+!)RfO**>F07$F\\o$\"3YG?hSf****>F07$$!3cpmmTF07$F`o$\"3O\"4q;3v.*>F07$$!3qnmmTF07$Fco$\"3^p28i5:f>F07$Ffo$\"3$o7=%z5#4\">F07$Fio$\"3W8sOI<, W=F07$F\\p$\"3U'4<\"*4O0x\"F07$F_p$\"3ET)G%oD#=o\"F07$Fcp$\"3#)pnMQ09$ e\"F07$Ffp$\"3!\\^o&)>(z%\\\"F07$Fip$\"3XthKG]a*R\"F07$F\\q$\"3]UP7,Zs .8F07$F_q$\"3'G@jf^/W@\"F07$Fbq$\"3MD2/t#*QL6F07$Feq$\"3'e.VJat00\"F07 $Fhq$\"3lV&o#4bmH)*F^o7$F[r$\"3'Ql.qF^o7$F]s$\"3@CkpTn>0nF^o7$F`s$\"3N'fI\"H(>vX'F^o7$Fcs $\"3W9a_5/oRiF^o7$Ffs$\"36b%*fj0dagF^o7$Fis$\"3lY'=66V\"3fF^o7$F\\t$\" 3F<:y9PZkdF^o7$F_t$\"3S8rToS*[l&F^o7$Fbt$\"3;(=X\\F^o7$F.$\"3!\\\">%oO AZ$\\F^o7$F2$\"3[@.;J-+C\\F^o7$F5$\"3k(=')>'y\")4\\F^o7$F8$\"3X(*R[wz& G*[F^o7$F;$\"38AF#zW*zs[F^o7$F>$\"3#z]!4`f%3&[F^o7$FA$\"31TN*pb!4C[F^o 7$FD$\"30_3RSpK\"z%F^o7$FG$\"3P7wK&Q$F^o7$Ffo$\"3$*o)f5<(*)yIF^o7$Fio$\"3.<\"y(*eLHp#F^o7$F\\p$\"36QUY kn7lAF^o7$F_p$\"3uq&*Q!fQfq\"F^o7$Fcp$\"3_@)znTpB!**Fhil7$Ffp$\"3wLD6c k*oD#Fhil7$Fip$!3q_D1^Z?\"*yFhil7$F\\q$!3%pG,.ApU5#F^o7$F_q$!3'*z;j'=x #QPF^o7$Fbq$!3\"=PMR3)yfdF^o7$Feq$!3=!fv&QWE4()F^o7$Fhq$!3L$4'4@;$fA\" F07$F[r$!3StP7jqWWx0f*eL7Fhdo7$F]s$!3Wd;tW>_7 BFhdo7$F`s$!3z\\^a)4MgJ&Fhdo7$$\"3KLLL)o))>v#F0$!3(H#z#ffEZl*Fhdo7$Fcs $!3_#zAmXZMC#Feeo7$$\"3CmmT:D*)yGF0$!3'*R$oqT833%Feeo7$$\"3#)****\\F'f 4#HF0$!3!HjpO#o-%f*Feeo7$$\"3)G$3_0kZJHF0$!3/Y9?KR#oF\"F]go7$$\"3Rm;a$ =$*>%HF0$!3sxLd()=<#y\"F]go7$$\"3!**\\i:'*4D&HF0$!3$e!*e'ybRfEF]go7$$ \"3SLLeRn-jHF0$!3*fMp`\"G5)Q%F]go7$$\"3\"p;/w^VN(HF0$!3(*\\$z$ok4r&)F] go7$$\"3(***\\i&HgS)HF0$!3VrzK#4:9O#Fbho7$$\"3s;aj%o=$*)HF0$!3x]ng'4$) )e_Fbho7$$\"3.Lektqd%*HF0$!3J*)p_<@CS?F`io7$$\"3y\\ilia$)**HF0$!3!H*)y [/oi@#!\"*7$Ffs$!3)RV+C6nBJ#F`io7$$\"3)H3_]wD*4IF0$!3C'[Dpz$)**3'Fbho7 $$\"3()*\\P%ywv9IF0$!3I\\v'[:f[v#Fbho7$$\"3v;H#=f*e>IF0$!3B\"HC&G$4Mc \"Fbho7$$\"3>L$3_]@W-$F0$!3EXj?\"\\>f+\"Fbho7$$\"3_m\"z>L&3MIF0$!3FPw3 'GcL;&F]go7$$\"3&)***\\(e\"\\P/$F0$!3mFXC![9Q8$F]go7$$\"3=L3_&)HT`IF0$ !3u?v^X&)3-@F]go7$$\"3^m;H7o2jIF0$!3`4%)pIj.2:F]go7$$\"3$)*\\i!R1usIF0 $!3:M%p,GeH8\"F]go7$$\"3;LL$eY/C3$F0$!3A;P(Hc!)e#))Feeo7$$\"3Zmm\"Hxf5 7$F0$!3a<-qh&QS3%Feeo7$Fis$!3Y\"e\\g)f7UBFeeo7$$\"3>LLLo$!3Jb[%R&HSJeF^o7$FA$!3)yF_Ec\" p/&)F^o7$FD$!3)3$z^X$y%=7F07$FG$!3wai\"3n([@Feeo7$Fcbs$!3&[W%=\">nEa$Feeo7$Fhbs$!3@%*>`P_x* =)Feeo7$$!3<+]PpKLj5F0$!33]a#o*z%[\\\"F]go7$F]cs$!34p)fR&*ela$F]go7$$! 3=+v=i:-I5F0$!3+`Kf;Q4cmF]go7$$!3$pm\"zfw\"*=5F0$!3+1r\"eo\\kn\"Fbho7$ $!3>]Pf3dO85F0$!3-#*H-G&p&eLFbho7$$!3oLeRdP\"y+\"F0$!3[&)))pBk8F)*Fbho 7$$!3%p\"z>1=E-5F0$!3NVR*\\fVG<\"Ffio7$F\\o$!3OU:whuoUbF`io7$$!3so\"zW Paz\"**F^o$!3!>vOaGQK\"*)Fbho7$$!3WN$e*)>5)o)*F^o$!3z)*F^o$!3r3EI&)*)*[%=Fbho7$$!3*)om\"z%=_q(*F^o$!3HTY6C*z#R6Fbh o7$$!3L-](o\\LAn*F^o$!3y.(pBO')Re&F]go7$$!3yNL$e9XRd*F^o$!3?'ooW1rVI$F ]go7$$!3n-+vV%otP*F^o$!3brc-sB#F^o7$Fjr$\"3[)ePbrTuo#F^o7 $F]s$\"3=N\"Rv/%4wIF^o7$F`s$\"3W&*\\b-Oh&Q$F^o7$Fcs$\"3Y.TS!fN'[OF^o7$ Ffs$\"3=()4[5:olQF^o7$Fis$\"3cy8\"=.KL.%F^o7$F\\t$\"3>_q*HlsW>%F^o7$F_ t$\"3_r1/#>3_J%F^o7$Fbt$\"3#)eY4,#4PU%F^o7$Fet$\"3,cl(H`m6^%F^o7$FhtFh _oFjt-F$6$7co7$F($\"3$*H*GCbs'fQF^o7$F.$\"30k*[]$GtKOF^o7$F2$\"3'HB(oZ H.(R$F^o7$F5$\"3%RqS#\\@C!3$F^o7$F8$\"3`**zslYu$p#F^o7$F;$\"3i`pL.?fDA F^o7$F>$\"3)*45j4%4\"*p\"F^o7$FA$\"3())=&\\?p(o.\"F^o7$FD$\"3#>S5:<&*) Q>Fhil7$FG$!3m*>NLx]q^)Fhil7$FJ$!34,d*zrGr?#F^o7$FM$!3*Qe.vSy3r$F^o7$F P$!3RT6TkLxreF^o7$FS$!3m;X1Jo8I()F^o7$FV$!3pboqFZ#>C\"F07$FY$!3:N?$oqG Ep\"F07$Ffn$!3#3C+(zV*yW#F07$Fin$!3UJcNAQ=$Q$F07$F\\o$!3;Fl:=rcj]F07$F `o$!3k4CdvVrJvF07$Fco$!3*fIb(3j6]7Fhdo7$Ffo$!3*pHEtazdF#Fhdo7$Fio$!3:Y Cr:Kvq`Fhdo7$Fihl$!3hG2CAY^8$*Fhdo7$F\\p$!3&)pJI0DSx>Feeo7$Fail$!3QP?* eiB-Z$Feeo7$Ffil$!3n>X(3D7-f(Feeo7$$!3vsm;/1binFhil$!3R#o^i6*)4J\"F]go 7$F\\jl$!3.Ga!\\zrey#F]go7$$!3TtmTN6ryNFhil$!3jT+DHX(Qo%F]go7$$!3'p++D JJu^#Fhil$!3\\>UXvZ^m%*F]go7$$!3tt;/,9z')>Fhil$!3V9!Gw\"4\"*>:Fbho7$$! 3^SLe*[^hX\"Fhil$!32l$G>E&eHGFbho7$$!3%G2]7y:^D*Fap$!3k5e#QrnX+(Fbho7$ F_p$!3\"eOI0eF![QF`io7$$\"3BTK3x19j:Fap$!3RAA0V[ebCFfio7$$\"3/BK$3-)* \\2(Fap$!3h6>=arl)>\"F`io7$$\"3\\?$ek`&oe7Fhil$!3(Rp#=\\r3(y$Fbho7$$\" 3n=L$3Fr)4=Fhil$!3@@1-kkgJ=Fbho7$$\"3.:LeRFC7HFhil$!3+#pcr$e]tqF]go7$F djl$!3:6\\VTyu@PF]go7$$\"3w2L3xc)p6&Fhil$!3mu4P@e^!H#F]go7$$\"37/L$e9d $>iFhil$!3*)e10oJ<]:F]go7$$\"3\\+Le9'Gv$e&)F^o7$Fdr$!3UXk@=7h;fF^o7$Fgr$!3vm#p7W'3#z$F^o7$Fjr$!3Nw @n2\"*)H<#F^o7$F]s$!3p]:ct6Po$)Fhil7$F`s$\"3UghU#Hj\"y=Fhil7$Fcs$\"3#y *=Z)[z<.\"F^o7$Ffs$\"3NhD*[^,,r\"F^o7$Fis$\"3$z?!)3hMIA#F^o7$F\\t$\"3k E?uSX82FF^o7$F_t$\"3(z'3df:>kIF^o7$Fbt$\"3A)4AMLv4Q$F^o7$Fet$\"3'y.%z+ R_LOF^o7$FhtFciuFhx-F$6$7ao7$F(Fd^s7$F.$!3nMKvnp'pt)F^o7$F2$!3S`A'y?PV ?\"F07$F5$!3PNR79\"))oq\"F07$F8$!3X`Bs;^lDCF07$F;$!3ea6I*G=m[$F07$F>$! 3M')4I`>L**\\F07$FA$!3oLU$[&yvhvF07$FD$!3z0qhYOqW7Fhdo7$FG$!3&fN')*=q \\4BFhdo7$FJ$!3)4LS6ws&ybFhdo7$Fdbm$!3]/2KWdBA'*Fhdo7$FM$!31BqtY(=J-#F eeo7$$!3!)****\\#f^'G@F0$!3msSEm=6:OFeeo7$F\\cm$!3BMe,q!4\\>)Feeo7$$!3 U++Dmc%R1#F0$!3oL&4[AMjY\"F]go7$$!3+nm;CqPU?F0$!3IaLt!p1,M$F]go7$$!3H+ ]7.FfJ?F0$!3>A\")32%G/,'F]go7$$!3eLL3#Q33-#F0$!3#e0k'\\sh&Q\"Fbho7$$!3 ++Dc@iT:?F0$!3W!)*yV>;X_#Fbho7$$!3(omT51C+,#F0$!3N'H5]iI6(fFbho7$$!3tL 3_+>j/?F0$!3dO$Rrx0mz#F`io7$FP$!3%3)o)Hoq!Q5!#57$$!32++vQa#Q*>F0$!3$p` KB'>vt:F`io7$$!3?++]P6T))>F0$!3-O\")HX@XnWFbho7$$!3M++DOo*H)>F0$!3^eA3 Q?Dv?Fbho7$$!3D+++NDex>F0$!39fE#)4$GQ>\"Fbho7$$!33++]KRvm>F0$!3:^>#zd& QFaF]go7$$!38+++I`#f&>F0$!3+\"o`:W\"o(3$F]go7$$!3++++D\"oU$>F0$!3EDNi0 1n(Q\"F]go7$Fdcm$!3Q;KsJIkYyFeeo7$$!33+++5lHp=F0$!3Ri4N6_=-NFeeo7$FS$! 35BT==wOr>Feeo7$F\\dm$!39l3_F%G!\\*)Fhdo7$FV$!3Ey<-.epg]Fhdo7$FY$!3%[x xRG-KP#Fhdo7$Ffn$!3)\\,iPaHiA\"Fhdo7$Fin$!3+$e9Q0>'=yF07$F\\o$!3@yWKWh !\\)\\F07$F`o$!3&fKE04%p'[$F07$Fco$!3O_yc%z+xT#F07$Ffo$!3wp+v@pRFWq@\"F07$F\\p$!3o`$[Nj8^s)F^o7$F_p$!3&HJ#[(y8u\"fF^o7$Fcp$ !3OMrvmB&Qq$F^o7$Ffp$!3$)=hkaY<*>#F^o7$Fip$!3RvME)\\>>-*Fhil7$F\\q$\"3 9?4q=\\0a;Fhil7$F_q$\"3U#)RSF%yV+\"F^o7$Fbq$\"3jhaX%3e`m\"F^o7$Feq$\"3 S[\\L)GUEE#F^o7$Fhq$\"3JggOagk,FF^o7$F[r$\"3?M?5g'H()3$F^o7$F^r$\"3IW. 42E>!Q$F^o7$Far$\"3ZN.$y\\(=YOF^o7$Fdr$\"3)*>/I<#e]&QF^o7$Fgr$\"3eg1'H G2!QSF^o7$Fjr$\"3G^HOR3'y=%F^o7$F]s$\"3v=7&*>6E>VF^o7$F`s$\"3>![y-q2`U %F^o7$Fcs$\"3Rv0$y4Fk^%F^o7$Ffs$\"3#[F^o7$FhtFf^rF^dl-F$6$7co7$F(Fcjt7$F.$!3R;$)G#>C(=yF0 7$F2$!3i>0n^xHA7Fhdo7$F5$!3[afs56cpAFhdo7$F8$!3a6$\\_>b*f`Fhdo7$F^\\n$ !3DYgz0)[@\")*Fhdo7$F;$!3?ipPKRRl]hIF0$!3@Jmme6-&e\"F]go7$$!3wLLL LbpTIF0$!3Wm]h\"Q;-X$F]go7$$!3*om;//!zJIF0$!3J\"z/8$z-OfF]go7$$!3-++]Z X)=-$F0$!3%)>duSPo_7Fbho7$$!3\"omT5!=$p,$F0$!3G339PHy#4#Fbho7$$!3fLLea !z>,$F0$!3c^[fm-:\"=%Fbho7$$!3%***\\73j-2IF0$!3f&RybTF`@\"F`io7$F>$!3a f,mH)eaR\"Ffio7$$!3dLLez_%p*HF0$!3Olt'4l]*HkF`io7$$!3'*****\\(*p\"=*HF 0$!3w>\\j[(o-'*)Fbho7$$!3\"om;ar)o')HF0$!3>9q\"*G**3'Q$Fbho7$$!3lLLLL/ c\")HF0$!37(>+1W9Xw\"Fbho7$$!30++D^@VwHF0$!3$4tO;*y6!3\"Fbho7$$!3*omm \"pQIrHF0$!3HN&Fhdo7$FG$!3[cS]3(*y\"H#Fhdo7$F J$!3$\\85tiK#=7Fhdo7$FM$!3&=y-\"fS*Qw(F07$FP$!3L]!*z(*R'\\,&F07$FS$!3B ^,E_?z&Q$F07$FV$!3t#QEWIR&yBF07$FY$!374BcGuGWz%R&)\\$QF^o7$Fc o$!3LQ.f`v5P@F^o7$Ffo$!3PhN%pJ(\\L')Fhil7$Fio$\"32ry&emk.(>Fhil7$F\\p$ \"3^cmN=/hU)*Fhil7$F_p$\"3eV_=&4sqn\"F^o7$Fcp$\"3_(*3!z6!>hAF^o7$Ffp$ \"3w.(fZ(\\)*zEF^o7$Fip$\"3-+N=d-qcIF^o7$F\\q$\"3qj*[#e7tyLF^o7$F_q$\" 3\"Rdm;#f(*ROF^o7$Fbq$\"3=H/'zl.7&QF^o7$Feq$\"3jSray\"3k/%F^o7$Fhq$\"3 bcL9=%HE>%F^o7$F[r$\"3.hZg)plNK%F^o7$F^r$\"3&pa=jZRMU%F^o7$Far$\"36%[U ]nub^%F^o7$Fdr$\"3Dw8+g&*f)e%F^o7$Fgr$\"3#[qhELgIl%F^o7$Fjr$\"38b=sgF@ 1ZF^o7$F]s$\"3b)e$pW\\3`ZF^o7$F`s$\"3$3I;'4f4\"z%F^o7$Fcs$\"3g%3mm<))Q #[F^o7$Ffs$\"3?sU-')zH^[F^o7$Fis$\"3Cr3u(G\"ps[F^o7$F\\t$\"3jN,&[QRM*[ F^o7$F_t$\"3(\\PE+81\"4\\F^o7$Fbt$\"3\"G]!p*4vK#\\F^o7$Fet$\"3J[TL-'eZ $\\F^o7$FhtFjisF]`m-F$6$7W7$F(Fgjm7$F.$\"3S-h`sv,w^F^o7$F2$\"3%\\x*G*) )eY?&F^o7$F5$\"3$=ynpEFCC&F^o7$F8$\"3j$oS&*f>uG&F^o7$F;$\"3L9dCd2RS`F^ o7$F>$\"3!oxX^=o!)R&F^o7$FA$\"3,+vRkM%zY&F^o7$FD$\"3#R2?j2!*Gb&F^o7$FG $\"3nan(GEiDl&F^o7$FJ$\"3qc&=,2hLx&F^o7$FM$\"3#zUCpWvv*eF^o7$FP$\"398G \")e$R01'F^o7$FS$\"32>SsH#>ED'F^o7$FV$\"3pTprf\"G)okF^o7$FY$\"3\"*=yHE ^L&p'F^o7$Ffn$\"3_\\w/j-82qF^o7$Fin$\"3JsiK1d\"=J(F^o7$F\\o$\"3()Rq`PE o=xF^o7$F`o$\"3WoH1#)=uI\")F^o7$Fco$\"3&)fnAjd7W')F^o7$Ffo$\"3;Vz%=XCp >*F^o7$Fio$\"3[Z+kR2pV)*F^o7$F\\p$\"3Fg=v^y?]5F07$F_p$\"3X9'*\\N)o!G6F 07$Fcp$\"3s9Y'RPGg@\"F07$Ffp$\"3\"fO\"eSWv(H\"F07$Fip$\"37!ze#G(=-R\"F 07$F\\q$\"3)e(y89J\\)[\"F07$F_q$\"3m\"G,2TK\\e\"F07$Fbq$\"3$[FoKF07$F^ r$\"31)eNw,S%e>F07$F_er$\"3hOY8/!=n(>F07$Far$\"3^[U$Q/[)*)>F07$F\\fr$ \"3\\EvS\")=L(*>F07$Fdr$\"3Eu*QtJ%****>F07$Fd[s$\"3>A!pB\"pe(*>F07$Fgr $\"39mz$3W3**)>F07$Fa\\s$\"3w$GK?Z3u(>F07$Fjr$\"3)f5'fe*Q,'>F07$F]s$\" 3@neP?^f4>F07$F`s$\"3MVK1IWtW=F07$Fcs$\"3Wl[P2+El*o n\"F07$Fis$\"3m]9!Qi$R!f\"F07$F\\t$\"3fW(HRvq%)[\"F07$F_t$\"3d(>(ooqZ( R\"F07$Fbt$\"30\"[H))G/II\"F07$Fet$\"3Sz]'*y$\\n@\"F07$FhtFc`mFijm-F$6 $7W7$F(Fcen7$F.$\"33`W5'y%*\\1&F^o7$F2$\"3#Q$pmXmhv]F^o7$F5$\"3&3/w.>V '*3&F^o7$F8$\"3%z1zo:#Q1^F^o7$F;$\"3$GK?p&48E^F^o7$F>$\"3dfRq?boZ^F^o7 $FA$\"3$erqJTqQ<&F^o7$FD$\"3]w'Hr#*4e?&F^o7$FG$\"3u:2(y$pVV_F^o7$FJ$\" 3+6X#*e)f#*G&F^o7$FM$\"3o@yr0LjO`F^o7$FP$\"3P9Pw1F=*R&F^o7$FS$\"3=7Th= k\\taF^o7$FV$\"3V5AThJ#zb&F^o7$FY$\"3'G:9*H[FZcF^o7$Ffn$\"3>!3)=$3D=x& F^o7$Fin$\"3sO2%p!)H`*eF^o7$F\\o$\"3$f(G%*\\)GJ1'F^o7$F`o$\"3$H>r3iplB 'F^o7$Fco$\"3D0cKXy&yX'F^o7$Ffo$\"3ym/JXV*Hq'F^o7$Fio$\"3X]SXk)H&**pF^ o7$F\\p$\"3#)\\$o&f<88tF^o7$F_p$\"3lu%z>#o#4q(F^o7$Fcp$\"3Kq]M?m(R;)F^ o7$Ffp$\"3tk6?a1g@')F^o7$Fip$\"3:!oX(48.x\"*F^o7$F\\q$\"3*eCcJ)3(4#)*F ^o7$F_q$\"3rR\\TMP;_5F07$Fbq$\"38t,^4caE6F07$Feq$\"3GN\"[7'3G;7F07$Fhq $\"3^h8sjq_-8F07$F[r$\"3Dxu^7-6*R\"F07$F^r$\"3aa%GUi()*)[\"F07$Far$\"3 !\\:S,OLue\"F07$Fdr$\"3'fU+4'yax;F07$Fgr$\"3e_A,(Q!)ew\"F07$Fjr$\"3;Ap u:80V=F07$F]s$\"3x#HcLa[/\">F07$F`s$\"3+$\\3V#)4#f>F07$Feat$\"3f(ofjJo r(>F07$Fcs$\"3jKHR6%f+*>F07$Fbbt$\"3FT6_p'fw*>F07$Ffs$\"30y_zp-****>F0 7$Feht$\"34J8;VkX(*>F07$Fis$\"3Elm\\#ou/*>F07$Fbit$\"3s:u^.<6x>F07$F\\ t$\"3i?X'pPA#e>F07$F_t$\"3%zfH-O%f6>F07$Fbt$\"3!3!Gl7?\"f%=F07$Fet$\"3 n%)Q\"Htq0x\"F07$FhtF_[nFeen-%+AXESLABELSG6$Q\"y6\"Q!6\"-%%VIEWG6$;F(F ht;$!+++++:Fcet$\"+++++DFcet" 1 2 0 1 10 0 2 9 1 4 2 1.000000 39.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1 8" }}}}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 33 "A partial differential polynomial" }}{PARA 0 "" 0 "" {TEXT -1 257 "In this example we will see that, just as the Rosenfeld_Groebner decomposition depends on the ranking, so does the minimal characteris tic decomposition. Nonetheless, the underlying prime decomposition of \+ the minimal characteristic decompositions is unique." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart;wi th(diffalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 786 "pp := expa nd(-y[s]*y[t]+y[s]^2+y[]*y[s,s]*y[s]*y[t]-2*y[s]*y[t]*y[]*y[s,t]+ y[s] *y[t]*y[]*y[t,t]+4*y[s]^2*y[s,s]*y[s,t]-2*y[s]^2*y[s,s]*y[t,t]-2*y[s]^ 2*y[s,s]*y[t]+3*y[s]*y[s,s]*y[t]^2+y[s,s]*y[t]^2*y[s,t]+y[s]*y[s,t]^2* y[t]+y[s]^2*y[s,t]*y[t,t]+4*y[s]^2*y[s,t]*y[t]+3*y[s]*y[s,t]^2*y[]-2*y [s]*y[s,t]*y[t]^2+y[s]^2*y[]*y[s,t]-y[s]^2*y[]*y[t,t]+y[]^2*y[s,s]*y[s ,t]-y[]*y[s,s]*y[t]^2-y[]^2*y[s,s]*y[t,t]-y[s]^2*y[t]*y[t,t]-y[]*y[s,t ]^2*y[t]+y[]*y[s,t]*y[t]^2+y[]^2*y[s,t]*y[t,t]-4*y[s]*y[s,s]*y[s,t]*y[ t]-4*y[s]*y[s,s]*y[]*y[s,t]+3*y[s]*y[s,s]*y[]*y[t,t]+y[s,s]*y[t]*y[s]* y[t,t]+2*y[s,s]*y[t]*y[]*y[s,t]-y[s,s]*y[t]*y[]*y[t,t]-2*y[s]*y[s,t]*y []*y[t,t]+y[s]^3*y[t]-2*y[s]^2*y[t]^2+y[s]*y[t]^3-y[]^2*y[s,t]^2-y[]*y [s]+y[]*y[t]-y[s,s]*y[t]^3-2*y[s]^2*y[s,t]^2-2*y[s]^3*y[s,t]+y[s]^3*y[ t,t]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ppp := subs(y[s,t ]=y[t,s],pp);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$pppG,^p**&%\"yG6#% \"sG\"\"\"&F(6#%\"tGF+&F(6\"F+&F(6$F.F.F+F+*&F'F+F,F+!\"\"**&F(6$F*F*F +F,F+F'F+F1F+F+**F6F+F,F+F/F+F1F+F4*$)F'\"\"#F+F+*(F:F+F/F+F1F+F4*(F/F +F6F+)F,F;F+F4*()F/F;F+F6F+F1F+F4*(F:F+F,F+F1F+F4**F/F+F6F+F'F+F,F+F+* &F'F+)F,\"\"$F+F+*&F/F+F'F+F4*&F/F+F,F+F+*&F6F+FDF+F4*&)F'FEF+F1F+F+*& F@F+)&F(6$F.F*F;F+F4*(F;F+F:F+FLF+F4*(F;F+FJF+FMF+F4*(F@F+FMF+F1F+F+*( F6F+F>F+FMF+F+*(F;F+F:F+F>F+F4*,F;F+F'F+F,F+F/F+FMF+F4*,\"\"%F+F'F+F6F +FMF+F,F+F4*,F;F+F6F+F,F+F/F+FMF+F+*,F;F+F'F+FMF+F/F+F1F+F4*,FEF+F'F+F 6F+F/F+F1F+F+**F;F+F:F+F6F+F1F+F4**F;F+F:F+F6F+F,F+F4**FEF+F'F+F6F+F>F +F+*&FJF+F,F+F+*(F'F+FLF+F,F+F+*(F:F+FMF+F1F+F+*(F:F+F/F+FMF+F+*(F@F+F 6F+FMF+F+**F;F+F'F+FMF+F>F+F4**FVF+F:F+FMF+F,F+F+**FVF+F:F+F6F+FMF+F+* *FEF+F'F+FLF+F/F+F+*,FVF+F'F+F6F+F/F+FMF+F4*(F/F+FLF+F,F+F4*(F/F+FMF+F >F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "p := evaln(p): ali as(`p`=p);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 26 "We choose a first ranking:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "alias( p = pp):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "R1 := differential_ring(rank ing=[y,z],derivations=[s,t]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "G1 := equations(Rosenfeld_Groebner([p],R1));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#G1G7%7#%\"pG7#,**&&%\"yG6\"\"\"\"&F,6#%\"sGF.!\"\" *&F+F.&F,6#%\"tGF.F.*&F/F.F4F.F2*$)F/\"\"#F.F.7$,&F/F.F+F2,&F4F.F+F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "According to the regular component theorem (Hubert 97), we know that \+ " }{XPPEDIT 18 0 "[y[s]-y,-y[]+y[t]]" "6#7$,&&%\"yG6#%\"sG\"\"\"F&!\" \",&&F&6\"F*&F&6#%\"tGF)" }{TEXT -1 14 " is redundant." }}{PARA 0 "" 0 "" {TEXT -1 41 "Let us study the other singular solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "a := y[s]^2-y[s]*y[t]-y[]*y[s]+y[]*y[t];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG,**&&%\"yG6\"\"\"\"&F(6#%\"sGF*!\"\"*&F'F*&F(6#%\"tGF*F**& F+F*F0F*F.*$)F+\"\"#F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "preparation_polynomial(p,z[]=a,R1);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,&*&&%\"zG6#%\"sG\"\"\"&F&6#%\"tGF)F)&F&6\"F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "By LPT (Hubert 97) \+ " }{XPPEDIT 18 0 "\{a\};" "6#<#%\"aG" }{TEXT -1 1 ":" }{XPPEDIT 18 0 " s[a] " "6#&%\"sG6#%\"aG" }{TEXT -1 32 " is an essential component ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "And, indeed:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "equations(essential_components(p,R1));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7$7#%\"pG7#,**&&%\"yG6\"\"\"\"&F*6#%\"sGF,!\"\"* &F)F,&F*6#%\"tGF,F,*&F-F,F2F,F0*$)F-\"\"#F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 21 "With another ranking: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"qG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "alias(q=ppp):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "R2:= differential_ring(ranking=[y,z ],derivations=[t,s]):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " G :=equations(Rosenfeld_Groebner([q],R2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG7%7#%\"qG7#,&&%\"yG6#%\"sG!\"\"&F+6#%\"tG\"\"\"7# ,&F*F2&F+6\"F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "M := equa tions(essential_components(q,R2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"MG7%7#%\"qG7#,&&%\"yG6#%\"sG\"\"\"&F+6\"!\"\"7#,&F*F1&F+6#%\"tGF. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 228 "This, in appearance is different from the previously obtained min imal characteristic decomposition. Except that the irreducible factors of the differential polynomials appearing in any minimal regular deco mposition are the same." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&&%\"yG6#%\"sG\"\"\"&F&6\"!\"\"F),&F%F)&F&6#%\"tGF,F )" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 19 "Universal equation " }} {PARA 0 "" 0 "" {TEXT -1 204 "This equation was brought out by R. Rube l. The solution of this 4th order, homogeneous differential equation o f degree 7 has the property that its solution can approximate piecewis ely any smooth function." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 203 "rubel := 3*y[x]^4*y[x$2]*y[x$4]^2-4*y[x]^4*y[x$3]^2*y[x$4]\n + 6*y[x]^3*y[x$2]^2*y[x$3]*y[x$4] + 24*y[x]^2*y[x$2]^4*y[x$4]\n -1 2*y[x]^3*y[x$2]*y[x$3]^3-29*y[x]^2*y[x$2]^3*y[x$3]^2+12*y[x$2]^7;\n" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&rubelG,0*()&%\"yG6#%\"xG\"\"%\"\" \"&F)6$F+F+F-)&F)6&F+F+F+F+\"\"#F-\"\"$**F,F-F'F-)&F)6%F+F+F+F3F-F1F-! \"\"*,\"\"'F-)F(F4F-)F.F3F-F7F-F1F-F-**\"#CF-)F(F3F-)F.F,F-F1F-F-**\"# 7F-F " 0 "" {MPLTEXT 1 0 55 "R := differential_ring(ranki ng=[y,z], derivations=[x]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "equations(Rosenfeld_Groebner([rubel], R));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7&7#,0*()&%\"yG6#%\"xG\"\"%\"\"\"&F)6$F+F+F-)&F)6&F+F+F +F+\"\"#F-\"\"$**F,F-F'F-)&F)6%F+F+F+F3F-F1F-!\"\"*,\"\"'F-)F(F4F-)F.F 3F-F7F-F1F-F-**\"#CF-)F(F3F-)F.F,F-F1F-F-**\"#7F-F " 0 "" {MPLTEXT 1 0 41 "equations(essentia l_components(rubel,R));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7$7#,0*()&% \"yG6#%\"xG\"\"%\"\"\"&F)6$F+F+F-)&F)6&F+F+F+F+\"\"#F-\"\"$**F,F-F'F-) &F)6%F+F+F+F3F-F1F-!\"\"*,\"\"'F-)F(F4F-)F.F3F-F7F-F1F-F-**\"#CF-)F(F3 F-)F.F,F-F1F-F-**\"#7F-F " 0 "" {MPLTEXT 1 0 67 "preparation_polynomial(3*y[x,x]^4+y[x]^2*y[x,x,x]^2, \+ z[]=y[x,x],R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)&%\"zG6\"\"\"% \"\"\"\"\"$*&)&%\"yG6#%\"xG\"\"#F*)&F'F0F2F*F*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 44 "preparation_polynomial(rubel, z[]=y[x,x],R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*$)&%\"zG6\"\"\"(\"\"\"\"#7**\"#CF *)&%\"yG6#%\"xG\"\"#F*)F&\"\"%F*&F'6$F2F2F*F***\"#HF*F.F*)F&\"\"$F*)&F 'F1F3F*!\"\"*,\"\"'F*)F/F;F*)F&F3F*F=F*F6F*F***F+F*FAF*F&F*)F=F;F*F>** F;F*)F/F5F*F&F*)F6F3F*F***F5F*FFF*F" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "The straight lines, sol utions of " }{XPPEDIT 18 0 "y[x,x] = 0;" "6#/&%\"yG6$%\"xGF'\"\"!" } {TEXT -1 118 " are limits of both the non-singular solutions and the e ssential singular solutions (the leading coefficient theorem)." }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 14 "Chazy equation" }}{PARA 0 "" 0 " " {TEXT -1 104 "This equation arose in Chazy's work to extend the Pain leve analysis to 3rd order differential equations." }}{PARA 0 "" 0 "" {TEXT -1 184 "In doing so he uncovered certain differential equations \+ whose non-singular solution have no movable singularity whereas one of the singular solution has. Here is an example (see Ince)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "chaz y := -(y[x,x]+y[]^3*y[x])^2 + (y[]*y[x])^2*(4*y[x]+y[]^4) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&chazyG,&*$),&&%\"yG6$%\"xGF,\"\"\"*&)&F*6 \"\"\"$F-&F*6#F,F-F-\"\"#F-!\"\"*()F0F5F-)F3F5F-,&F3\"\"%*$)F0F;F-F-F- F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "R := differential_rin g(ranking=[y], derivations=[x]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "equations(Rosenfeld_Groebner([chazy],R));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%7#,(*$)&%\"yG6$%\"xGF+\"\"#\"\"\"F-**F,F-F (F-)&F)6\"\"\"$F-&F)6#F+F-F-*(\"\"%F-)F0F,F-)F3F2F-!\"\"7#,&*$)F3F,F-F 6*&F3F-)F0F6F-F-7#F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "equ ations(essential_components(chazy,R));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7#,(*$)&%\"yG6$%\"xGF+\"\"#\"\"\"F-**F,F-F(F-)&F)6\"\"\"$F-&F) 6#F+F-F-*(\"\"%F-)F0F,F-)F3F2F-!\"\"7#,&F3F6*$)F0F6F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The compo nent " }{XPPEDIT 18 0 "[y];" "6#7#%\"yG" }{TEXT -1 89 " appears as red undant, as well as a prime factor of the other singular regular compon ent " }{XPPEDIT 18 0 "[4*y[x]^2+y[x]*y[]^4];" "6#7#,&*&\"\"%\"\"\"*$&% \"yG6#%\"xG\"\"#F'F'*&&F*6#F,F'*$&F*6\"F&F'F'" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Here, our conclusions are:" }}{PARA 0 "" 0 "" {TEXT -1 20 "- the solutions of \+ " }{XPPEDIT 18 0 "4*y[x]+y^4 = 0;" "6#/,&*&\"\"%\"\"\"&%\"yG6#%\"xGF'F '*$F)F&F'\"\"!" }{TEXT -1 44 " are envelopes of the non-singular solut ions" }}{PARA 0 "" 0 "" {TEXT -1 19 "- the solution of " }{XPPEDIT 18 0 "y[x] = 0;" "6#/&%\"yG6#%\"xG\"\"!" }{TEXT -1 38 " are limits of \+ non-singular solutions." }}{PARA 0 "" 0 "" {TEXT -1 2 "- " }{XPPEDIT 18 0 "y = 0;" "6#/%\"yG\"\"!" }{TEXT -1 63 " is a limit of both the n on-singular solution of the equation " }{XPPEDIT 18 0 "chazy;" "6#%&ch azyG" }{TEXT -1 36 " and the non-singular solutions of " }{XPPEDIT 18 0 "4*y[x]+y^4 = 0;" "6#/,&*&\"\"%\"\"\"&%\"yG6#%\"xGF'F'*$F)F&F'\" \"!" }{TEXT -1 35 ", the essential singular component." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "ode2 := jet2diff(chazy, x);\node1 \+ := 4*diff(y(x),x)^2+diff(y(x),x)*y(x)^4;\nsingsol := RootOf(3*_Z^3*(x- C)-4);\nodetest(y(x)=singsol, ode1);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%%ode2G,&*$),&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F/\"\"#\"\"\"*&)F,\"\" $F4-F*6$F,F/F4F4F3F4!\"\"*()F,F3F4)F8F3F4,&F8\"\"%*$)F,F?F4F4F4F4" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ode1G,&*$)-%%diffG6$-%\"yG6#%\"xGF. \"\"#\"\"\"\"\"%*&F(F0)F+F1F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%( singsolG-%'RootOfG6#,&*&,&%\"xG!\"$*&\"\"$\"\"\"%\"CGF/F/F/)%#_ZGF.F/F /\"\"%F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 120 "plot( \{seq(C/3, C=-2..2), seq( (4/(3*(x-2* C)))^1/3, C=-2..1)\}, x=-3..3, -1..1, title=\"singular solutions\",dis cont=true);" }}{PARA 13 "" 1 "" {GLPLOT2D 356 236 236 {PLOTDATA 2 "6.- %'CURVESG6$7S7$$!\"$\"\"!$F*F*7$$!3*)*****\\2<#pG!#iUC\"F/F+7$$!3-++Dhka I6F/F+7$$!3s******\\XF`**!#=F+7$$!3u*******>#z2))FgnF+7$$!3R++]7RKvuFg nF+7$$!3t,+++P'eH'FgnF+7$$!3q)***\\7*3=+&FgnF+7$$!3[)***\\PFcpPFgnF+7$ $!3:)****\\7VQ[#FgnF+7$$!32)***\\i6:.8FgnF+7$$!3Xb+++v`hH!#?F+7$$\"3_* ***\\(QIKH\"FgnF+7$$\"38****\\7:xWCFgnF+7$$\"3F,++vuY)o$FgnF+7$$\"3*y* *****4FL(\\FgnF+7$$\"3@)****\\d6.B'FgnF+7$$\"3s****\\(o3lW(FgnF+7$$\"3 7*****\\A))oz)FgnF+7$$\"3e******Hk-,5F/F+7$$\"36+++D-eI6F/F+7$$\"3t*** \\(=_(zC\"F/F+7$$\"3M+++b*=jP\"F/F+7$$\"3f***\\(3/3(\\\"F/F+7$$\"33++v B4JB;F/F+7$$\"3u*****\\KCnu\"F/F+7$$\"3s***\\(=n#f(=F/F+7$$\"3P+++!)RO +?F/F+7$$\"30++]_!>w7#F/F+7$$\"3N++v)Q?QD#F/F+7$$\"3G+++5jypBF/F+7$$\" 3<++]Ujp-DF/F+7$$\"3++++gEd@EF/F+7$$\"39++v3'>$[FF/F+7$$\"38++D6EjpGF/ F+7$$\"\"$F*F+-%'COLOURG6&%$RGBG$\"*++++\"!\")F+F+-F$6$7S7$F($!3:LLLLL LLLFgn7$F-Fdu7$F1Fdu7$F4Fdu7$F7Fdu7$F:Fdu7$F=Fdu7$F@Fdu7$FCFdu7$FFFdu7 $FIFdu7$FLFdu7$FOFdu7$FRFdu7$FUFdu7$FXFdu7$FenFdu7$FinFdu7$F\\oFdu7$F_ oFdu7$FboFdu7$FeoFdu7$FhoFdu7$F[pFdu7$F^pFdu7$FbpFdu7$FepFdu7$FhpFdu7$ F[qFdu7$F^qFdu7$FaqFdu7$FdqFdu7$FgqFdu7$FjqFdu7$F]rFdu7$F`rFdu7$FcrFdu 7$FfrFdu7$FirFdu7$F\\sFdu7$F_sFdu7$FbsFdu7$FesFdu7$FhsFdu7$F[tFdu7$F^t Fdu7$FatFdu7$FdtFdu7$FgtFdu-Fjt6&F\\uF+F]uF+-F$6$7S7$F($\"3:LLLLLLLLFg n7$F-F\\y7$F1F\\y7$F4F\\y7$F7F\\y7$F:F\\y7$F=F\\y7$F@F\\y7$FCF\\y7$FFF \\y7$FIF\\y7$FLF\\y7$FOF\\y7$FRF\\y7$FUF\\y7$FXF\\y7$FenF\\y7$FinF\\y7 $F\\oF\\y7$F_oF\\y7$FboF\\y7$FeoF\\y7$FhoF\\y7$F[pF\\y7$F^pF\\y7$FbpF \\y7$FepF\\y7$FhpF\\y7$F[qF\\y7$F^qF\\y7$FaqF\\y7$FdqF\\y7$FgqF\\y7$Fj qF\\y7$F]rF\\y7$F`rF\\y7$FcrF\\y7$FfrF\\y7$FirF\\y7$F\\sF\\y7$F_sF\\y7 $FbsF\\y7$FesF\\y7$FhsF\\y7$F[tF\\y7$F^tF\\y7$FatF\\y7$FdtF\\y7$FgtF\\ y-Fjt6&F\\uF]uF]uF+-F$6$7S7$F($!3ImmmmmmmmFgn7$F-Fd\\l7$F1Fd\\l7$F4Fd \\l7$F7Fd\\l7$F:Fd\\l7$F=Fd\\l7$F@Fd\\l7$FCFd\\l7$FFFd\\l7$FIFd\\l7$FL Fd\\l7$FOFd\\l7$FRFd\\l7$FUFd\\l7$FXFd\\l7$FenFd\\l7$FinFd\\l7$F\\oFd \\l7$F_oFd\\l7$FboFd\\l7$FeoFd\\l7$FhoFd\\l7$F[pFd\\l7$F^pFd\\l7$FbpFd \\l7$FepFd\\l7$FhpFd\\l7$F[qFd\\l7$F^qFd\\l7$FaqFd\\l7$FdqFd\\l7$FgqFd \\l7$FjqFd\\l7$F]rFd\\l7$F`rFd\\l7$FcrFd\\l7$FfrFd\\l7$FirFd\\l7$F\\sF d\\l7$F_sFd\\l7$FbsFd\\l7$FesFd\\l7$FhsFd\\l7$F[tFd\\l7$F^tFd\\l7$FatF d\\l7$FdtFd\\l7$FgtFd\\l-Fjt6&F\\uF+F+F]u-F$6$7S7$F($\"3ImmmmmmmmFgn7$ F-F\\`l7$F1F\\`l7$F4F\\`l7$F7F\\`l7$F:F\\`l7$F=F\\`l7$F@F\\`l7$FCF\\`l 7$FFF\\`l7$FIF\\`l7$FLF\\`l7$FOF\\`l7$FRF\\`l7$FUF\\`l7$FXF\\`l7$FenF \\`l7$FinF\\`l7$F\\oF\\`l7$F_oF\\`l7$FboF\\`l7$FeoF\\`l7$FhoF\\`l7$F[p F\\`l7$F^pF\\`l7$FbpF\\`l7$FepF\\`l7$FhpF\\`l7$F[qF\\`l7$F^qF\\`l7$Faq F\\`l7$FdqF\\`l7$FgqF\\`l7$FjqF\\`l7$F]rF\\`l7$F`rF\\`l7$FcrF\\`l7$Ffr F\\`l7$FirF\\`l7$F\\sF\\`l7$F_sF\\`l7$FbsF\\`l7$FesF\\`l7$FhsF\\`l7$F[ tF\\`l7$F^tF\\`l7$FatF\\`l7$FdtF\\`l7$FgtF\\`l-Fjt6&F\\uF]uF+F]u-F$6$7 U7$F($\"3?WWWWWWWWFgn7$$!3&*****\\P&3Y$HF/$\"3^#*Q>&*QlrTFgn7$F-$\"3VY ?(QZ8/$RFgn7$$!3')**\\78.K7GF/$\"3mP#H!=Q7UPFgn7$F1$\"3D$[AFx\\5d$Fgn7 $F4$\"3p80>*)[4QKFgn7$F7$\"3S8oQ>*e-'HFgn7$F:$\"3t.Km0wNFFFgn7$F=$\"3l Zj+&[T>a#Fgn7$F@$\"3HGQ!=RtZP#Fgn7$FC$\"3?jz9'=VNA#Fgn7$FF$\"3#pYC=`A3 4#Fgn7$FI$\"34F;;9u()p>Fgn7$FL$\"3ZJ\\')3pPu=Fgn7$FO$\"3oKfB)Rstx\"Fgn 7$FR$\"3/')*fW*Hd*o\"Fgn7$FU$\"3tD!)>dlz7;Fgn7$FX$\"3!o^\"pG=))[:Fgn7$ Fen$\"3,J$>,kx\"z9Fgn7$Fin$\"3Ybxg#Gd[U\"Fgn7$F\\o$\"3*4x:_%Q[m8Fgn7$F _o$\"3wg:E$Rk'=8Fgn7$Fbo$\"3OX=r-p!*p7Fgn7$Feo$\"30n,L^crE7Fgn7$Fho$\" 3wl81oZn%=\"Fgn7$F[p$\"3LqJOw(G&[6Fgn7$F^p$\"3^VY%*pV$>6\"Fgn7$Fbp$\"3 fo1@\"48j2\"Fgn7$Fep$\"3E=$=+N7r/\"Fgn7$Fhp$\"3K)Hl1\"QI<5Fgn7$F[q$\"3 2IGH;3S#))*!#>7$F^q$\"3tl(H57-Ph*Fdil7$Faq$\"30&Q=:%RFn$*Fdil7$Fdq$\"3 0Kci1*[!3\"*Fdil7$Fgq$\"3eah'*)\\kq)))Fdil7$Fjq$\"3&yAU7haEm)Fdil7$F]r $\"3r!='oyY()o%)Fdil7$F`r$\"3tdfI3`qm#)Fdil7$Fcr$\"3<*[$*H$**4&3)Fdil7 $Ffr$\"3c-t-lyg.zFdil7$Fir$\"3[:u!GXuQt(Fdil7$F\\s$\"3hweR3(=Qc(Fdil7$ F_s$\"31<1kw!epS(Fdil7$Fbs$\"3L[6W%QMJD(Fdil7$Fes$\"3-8=28qw1rFdil7$Fh s$\"3Z(y\"y**QQxpFdil7$F[t$\"3:sx\\C;xMoFdil7$F^t$\"3dw%))4xn?r'Fdil7$ Fat$\"3'QF3Vq,ge'Fdil7$Fdt$\"31aU=HuppkFdil7$Fgt$\"3&)[j?\\j?\\jFdil-F jt6&F\\uF+F]uF]u-F$6%7gn7$F($!3?WWWWWWWWFgn7$$!36;T'f%G?yHF/$!3=@/+k$z Ma%Fgn7$$!3+ViyesBfHF/$!3DbI_(*3JLYFgn7$$!3g#)eXs%3z$HF/$!3@41%33w'QZF gn7$$!3f.Z30$Qk\"HF/$!3*\\exB,$p\\[Fgn7$$!3jX7n\"=q]*GF/$!3XJ(RdMqa'\\ Fgn7$$!3:yn&f>f_(GF/$!3[T+N&*3'y2&Fgn7$$!3^BlunguaGF/$!3g22QQ\\s*>&Fgn 7$$!3LXr?h8`LGF/$!3e\")3/ci1K`Fgn7$$!3&>p+0p%Q7GF/$!3MgPV\"*='3Z&Fgn7$ $!38qGUSFj!z#F/$!3$=HfOnw8i&Fgn7$$!3u\"\\a4ht9x#F/$!3jmQ0?0)4w&Fgn7$$! 3Nq0+p\\!*\\FF/$!3L\\fH$)onEfFgn7$$!3i808kxCGFF/$!3k5Nai!HH5'Fgn7$$!3& ztbU.xtq#F/$!3\")*[@)yA*HG'Fgn7$$!3$pyx?TC%)o#F/$!3Whu-xd'fX'Fgn7$$!3R R8<6z)em#F/$!3Q!eg\"HLYumFgn7$$!3h()eyQlzYEF/$!3?*z,T@s9(oFgn7$$!3Snu7 M()eCEF/$!3F.m$R[$z:rFgn7$$!3>!3PS1J\\g#F/$!30A&H?JEqM(Fgn7$$!3A>[P%[j Le#F/$!3$)\\,))HQl=wFgn7$$!3hV![#[g#Gc#F/$!3ax)R*Qtl'*yFgn7$$!3J#Go\") Q(RTDF/$!3)>22bC4#4#)Fgn7$$!3\\^8\"*)=><_#F/$!3(R%pT>T%)=&)Fgn7$$!3'>P !e#f$\\+DF/$!3))[%oOcA,)))Fgn7$$!3AcwLjhWyCF/$!3c_\" *eP#F/$!3t)>#z1UP#=\"F/7$$!35*oH,?&Q`BF/$!3tpUEankU\"F/7$$!3o3jXMT+#H#F/$!3# Q$*)Qk%[?_\"F/7$$!35>j(=^81F#F/$!3iHk*)e'eBk\"F/7$$!3PZ(QIg'[]AF/$!3Z% \\l:@CVx\"F/7$$!3\"4J)*R=[%HAF/$!3%Qm/[V9q$>F/7$$!31\"R8RGz)3AF/$!3wDp W6uvF@F/7$$!3#o#4v^bM(=#F/$!3xNT*4^CBP#F/7$$!35(pL$3ggm@F/$!3P&Q[LbPwm #F/7$$!3D_4rHoRX@F/$!3+wLDo%on0$F/7$$!3&p[&ztKOC@F/$!3Tkrn1'fPd$F/7$$! 3?'yp.iN]5#F/$!3=b$=kxo8B%F/7$$!3'GO>%[R)G3#F/$!3%3BuQ!*\\AO&F/7$$!3-1 \\&*G72j?F/$!3quap&e1n/(F/7$$!3d>LiPn%>/#F/$!3w;Wx@Zaf5!#;7$$!3V0?F/$!363i\"*=j*>=)F_\\n 7$$!3j$*R(o)R2/?F/$!3xK>&3WK44\"!#:7$$!3DH$\\K*fr-?F/$!39n$oiX(RO;Fc^n 7$$!3%o*pV'*p.-?F/$!3+bzMo;'==#Fc^n7$$!3VkYi**zN,?F/$!3\\kA]#o(ysKFc^n 7$$!3YKB\"G+z1+#F/$!3*=.a]WYba'Fc^n7$$!30+++1+++?F/$!39F/$\"39F/$\"3+$R9b3f\\ I\"Fc^n7$$!3h!4JLQ)=$*>F/$\"3j5f#\\<,[_'F_\\n7$$!3$ej'*zd#y*)>F/$\"39t =9F(o)\\VF_\\n7$$!3F\"=iExwj)>F/$\"3%*RLIC?SiKF_\\n7$$!3qrK*>;l&z>F/$ \"3kWp;-]$\\<#F_\\n7$$!3OiVK^Nvs>F/$\"3j.?Or8?J;F_\\n7$$!3YVl)*H.8f>F/ $\"3UWDtgwY(3\"F_\\n7$$!3yC(['3r]X>F/$\"3\")\\h$[v2g:)F/7$$!3?(3tfmg#= >F/$\"3q!\\&y-(QtV&F/7$$!3h\\uHBU,\"*=F/$\"3pWY#>5/!ySF/7$$!3bz^7b-gV= F/$\"3Sy)=A.@<%GF/7$$!3s4H&pG'='z\"F/$\"3='=3f`S1=#F/7$$!3m:#*yaBa*o\" F/$\"3&fC?JNy:V\"F/7$$!3wO!=u^\">#e\"F/$\"3[fKB1;vj5F/7$$!3M7z$)**3Nv9 F/$\"3;\\p$))\\q7Z)Fgn7$$!3o6,zqfHw8F/$\"3n*>'RhFgn7$$!3Xy//'zcw;\"F/$\"3'4f<%*px'R`Fgn7$$!3y\"p+?WB>1 \"F/$\"3>z3))**p#yt%Fgn7$$!38p`*3\"pjJ&*Fgn$\"3]w@%3Q'fXUFgn7$$!35]#yG V!ot&)Fgn$\"3pL0J)Hb'*)QFgn7$$!3c,d+I$[_\\(Fgn$\"3]8>,$[/Ub$Fgn7$$!3rL ^I\"3)Q7kFgn$\"3tA!H%HL&4F$Fgn7$$!3#oPdDe^)o`Fgn$\"3>))3GE$fw.$Fgn7$$! 3E-76p/A@WFgn$\"3uO3)ze$)G&GFgn7$$!3Me+Q=aR%H$Fgn$\"3:'H'\\@6XgEFgn7$$ !3KTb_%zE)RBFgn$\"3>'30cF[m^#Fgn7$$!3uwYFclVH7Fgn$\"3@(3AGftxO#Fgn7$$! 3i@ZPq/`lCFdil$\"3v_OwL*e*\\AFgn7$$\"3)>4=DTf#=$)Fdil$\"3keDv(pe=Fgn$\"3gH(*y58EL?Fgn7$$\"3,40lh28IHFgn$\"3nRUon^DQ>Fgn7 $$\"3s=)>#H2/9RFgn$\"3:#e*z_$3&e=Fgn7$$\"3AZH'3b?`(\\Fgn$\"3rw+-&\\M&z Qk\"Fgn7$$\"34o[9AHst!)Fgn$\"3'fsu$eK8$e\"Fgn7$$\"34qm-v#RW9*Fgn$\"3W* )QFx;(\\_\"Fgn7$$\"3bopiLE>>5F/$\"3#G,%>()Q1s9Fgn7$$\"32\\.)GRU07\"F/$ \"3\\7;qEQDC9Fgn7$$\"3/6.(3-uIB\"F/$\"3Wurv&p!ou8Fgn7$$\"3vRcm$p)=M8F/ $\"3#G%[(pF\"*HL\"Fgn7$$\"3_>%p$*=]@W\"F/$\"3`Im(Qg#=\"H\"Fgn7$$\"3&yN 52Nz*R:F/$\"3c#=:W5+bD\"Fgn7$$\"3L\"oBTYKpk\"F/$\"3vcVO;-o=7Fgn7$$\"3g &e%z3qcZag.c&>F/$\"3A_u(p&=eB6Fgn7$$\"3gs!\\nEsK1#F/$\"3ns\"*zk\"4Q4\"Fgn7 $$\"3GpHL%)*pp;#F/$\"3Ygi*H-*em5Fgn7$$\"3\"Rrbz(e,tAF/$\"3s[c_p)=,/\"F gn7$$\"3\"o%y.eO=yBF/$\"3%o8s%4V8:5Fgn7$$\"3I8-jD>#[Z#F/$\"3'z\"o5fY6K **Fdil7$$\"3\\qR\"fG!e&e#F/$\"3$f`XK9=Ap*Fdil7$$\"3bgFgn7$$!3Rq0+/\\r\\AF/$!39jStj*eb(>Fgn7$$ !3T808*GVZ=#F/$!3f))4b!p4V.#Fgn7$$!3%ytb#*4J@7#F/$!3QPE2j2L%4#Fgn7$$!3 Q`W\\KKFl?F/$!3'>QQif))>:#Fgn7$$!3ksY]HPm(*>F/$!3[[a=96#[A#Fgn7$$!3z?# >@h*QS>F/$!3?5r.V2\\!H#Fgn7$$!3znu(y>mP(=F/$!3vI[,n6$>P#Fgn7$$!3t8/P(= $z9=F/$!3\"46uqx3!\\CFgn7$$!3=>[7[/4]Aj#Fgn7$$!3t:;+f@>C;F/$!3J8u5\"4.kt#Fgn7$$!3%[o%*4 cd^c\"F/$!3wUhxP\"F/$!3ik[(GtWeA$Fgn7$$!3pY)='Hmd:8F /$!3CtFagUKyLFgn7$$!3(HL(\\[OL^7F/$!3\"fnF&4hw^NFgn7$$!3IJI([U%[)=\"F/ $!36!f^\"y**eRPFgn7$$!30^'p$pXnF6F/$!31r#\\@QZ7%RFgn7$$!31*oHEfb,1\"F/ $!3S_x*[vcA>%Fgn7$$!3Ko-,!*y'[***Fgn$!3\"f3@MbEnW%Fgn7$$!3q-eI;*)4Z$*F gn$!3FG\">&[=*[v%Fgn7$$!3Ua(H([R7g()Fgn$!3J99]I\\\\t]Fgn7$$!3\"p=j(o_S =\")Fgn$!3xZ-,A*GXZ&Fgn7$$!3!p!3A,!)f9vFgn$!3#)*yS$4TT9fFgn7$$!3G5J[Fa W$)oFgn$!32)4;%p[rckFgn7$$!3vWsYA%yjE'Fgn$!3PPNF7[_#4(Fgn7$$!37p#4]Xm. i&Fgn$!3\"y<(=2&[x!zFgn7$$!3NRO+],=)*\\Fgn$!3L[\"[dND@*))Fgn7$$!3#>>w( )y/>O%Fgn$!3yLpqYG#*=5F/7$$!3;.#)y3\")*3t$Fgn$!3_$)fxNKD\">\"F/7$$!3^i yp.&o5:$Fgn$!3:Px(oIc/T\"F/7$$!3Nkp_U$=l[#Fgn$!3AYit8nT(y\"F/7$$!3&)Hd @cn8#*=Fgn$!3))[W![L-*[BF/7$$!3F&>LP,-%e7Fgn$!3e;8\">1;=`$F/7$$!3/7w*3 3&>^&*Fdil$!3ce+&*=iG`YF/7$$!3TrKYC+P=lFdil$!3ldm?UrL=oF/7$$!3[`uMowx) )[Fdil$!38c0n5f6\"4*F/7$$!3bN;B7`=fKFdil$!3P&4>HInOO\"F_\\n7$$!3eEPA7Fdil$!3@A[OD]WOOF_\\n7$$!3B)3z0ynz9)F`p$!3$[lW\"*>mYX&F_\\n7 $$!3#eJMaLx46'F`p$!3yWOK![')GF(F_\\n7$$!3FW&*G!*o)R2%F`p$!32s\")[OC$44 \"Fc^n7$$!3HsZ9Xk*p.#F`p$!3.3/if;'==#Fc^n7$$!3'***************f!#E$!3( fS2uS2uS(F]`n7gn7$$\"3'***************fFjdp$\"3(fS2uS2uS(F]`n7$$\"3NL! Q!*>*[V?F`p$\"3j9ibc#H\\<#Fc^n7$$\"3Qmg2)Rsp3%F`p$\"34IDwCiY(3\"Fc^n7$ $\"3U*49rfb/8'F`p$\"3K2.]z$y(\\sF_\\n7$$\"3XK@:'zQR<)F`p$\"3X'\\K]6NtV &F_\\n7$$\"3&)>GU>04E7Fdil$\"3TUpij4*[i$F_\\n7$$\"3YE/BfryM;Fdil$\"3e3 BKb&o'=FF_\\n7$$\"3-Sc%)Q/=_CFdil$\"3+%*HGDfW7=F_\\n7$$\"3B`3Y=PdpKFdi l$\"3^1G6FXLf8F_\\n7$$\"3O!G\"px-O/\\Fdil$\"3v$=*3N2Bi!*F/7$$\"3y1<#p$ o9RlFdil$\"3aML>fKn'z'F/7$$\"3jI?)\\#\\)RQ*Fdil$\"3,%4BIg,it%F/7$$\"3X NUI,B)GA\"Fgn$\"3`P*fZ#3SMOF/7$$\"363Xx$*eui=Fgn$\"3m,Y.$*Q'fQ#F/7$$\" 3^)H'[<4&o]#Fgn$\"3bBp;G$>Hx\"F/7$$\"3P5UXAY*y9$Fgn$\"3&R'z$oSy=T\"F/7 $$\"39^bE'>CAu$Fgn$\"3RygcHyk(=\"F/7$$\"3ZjZ.X!=wN%Fgn$\"3FSRK(>D*>5F/ 7$$\"3P\")=wV#fS*\\Fgn$\"3)Q)Rr'zi%**))Fgn7$$\"3i\"3$\\n$f%GcFgn$\"3@& R\"y0$yj*yFgn7$$\"30lzVsy,\"G'Fgn$\"30aMpYR*f2(Fgn7$$\"3^;%)ye5&G](Fgn$\"33tN9CTnBfFgn7$$\"3Jl[porc_ \")Fgn$\"3ysUpq))e^aFgn7$$\"37AEWn!*oy()Fgn$\"3XJ1,Mbwi]Fgn7$$\"3\\ka0 NxEZ$*Fgn$\"3s([j:(f![v%Fgn7$$\"3UF`\\wiL-5F/$\"3j&4Ca>&3MWFgn7$$\"3Fz 2)QR5'f5F/$\"3a'RIly8W>%Fgn7$$\"3FKD73QBE6F/$\"3))\\w6;$*GYRFgn7$$\"3K 'eH'=o?&=\"F/$\"3VE#z;bJ*\\PFgn7$$\"3)3=vyb4*\\7F/$\"3pE*p<$G\"eb$Fgn7 $$\"3Nc>]m=_68F/$\"3j.g#y%)o()Q$Fgn7$$\"3K%Q)*p%y!eP\"F/$\"3^AGUw_UIKF gn7$$\"3@:`+XC%[V\"F/$\"3\\n&H'=R^(4$Fgn7$$\"3ThHDM#>&)\\\"F/$\"3IfglA 3*e'HFgn7$$\"3,xcCA:mk:F/$\"3E&=V#\\\\^SGFgn7$$\"3)G_!Qy&QAi\"F/$\"3Ly X@/%)pRFFgn7$$\"3O`6QwLU%o\"F/$\"3m@f(*GabQEFgn7$$\"33nE]djm[;a#Fgn7$$\"3vop7\"e::\"=F/$\"3/:Bq5)RMX#Fgn7$$\"3+\\.jOaKs=F/$\"3 &yF#pwjvtBFgn7$$\"3*4JqLTW)R>F/$\"3#3\"p#>\\M6H#Fgn7$$\"3At*)*p@80+#F/ $\"3)yU.u7_;A#Fgn7$$\"3y>%pV6!Hl?F/$\"3'))*=o05(>:#Fgn7$$\"3hCq76w)R7# F/$\"3e$)[Lt,]#4#Fgn7$$\"3O\"oB\"z%f\")=#F/$\"3')GxL$pL6.#Fgn7$$\"3O>z (e?S&[AF/$\"3%3H%3\\5fw>Fgn7$$\"3-*o^KYb;J#F/$\"3W^8+uSiA>Fgn7$$\"3evK vj@OtBF/$\"3i3-!zUOE(=Fgn7$$\"39t!*\\gL'zV#F/$\"3,s!HVF:I#=Fgn7$$\"37O '**4*>=+DF/$\"3IMhPr$[wx\"Fgn7$$\"3'3QAr_4Qc#F/$\"3yoM(*[9`L5pi#F/$\"3#4m%f:0*=p\"Fgn7$$\"3!Q@Ic:$*[o#F/$\"35[!HIW_`l\"Fgn7$ $\"3i.tur\"[8v#F/$\"39KTU!pp`h\"Fgn7$$\"32F%y.L'y5GF/$\"30scc_.@\"e\"F gn7$$\"3`!oEY!)fT(GF/$\"3t7GuEcMY:Fgn7$$\"3Mn`v0j\"[$HF/$\"3*o/hS!fQ9: Fgn7$Fgt$\"3S\"[\"[\"[\"[\"[\"FgnFfx-F$6%7gn7$F($!3%y)))))))))))))))Fd il7$$!3n\\uHHU,\"*GF/$!3'=ws*G(ep3*Fdil7$$!3**4H&HH'='z#F/$!3(zFOpz@mE *Fdil7$$!3r:#*ygBa*o#F/$!3`QGpk@Nx%*Fdil7$$!3#o.=M_\">#e#F/$!3j&G+!HgQ *p*Fdil7$$!3S7z$e!4NvCF/$!3&>onqpS4$**Fdil7$$!3t6,zwfHwBF/$!3+^Zwz@d:5 Fgn7$$!3YBlCN.ttAF/$!3YFW_))\\%*R5Fgn7$$!3]y//-oln@F/$!3Bu/B_KTm5Fgn7$ $!3$=p+![M#>1#F/$!3jv'*\\zB<%4\"Fgn7$$!3(p`*3(pjJ&>F/$!3!)44;O`FC6Fgn7 $$!31DyG\\!ot&=F/$!3IM!\\c5'>_6Fgn7$$!3@q0+R[_\\\"H\"Fgn7$$!3*e+QyaR%H8F/$!3p*4Z!pE*[L\"Fgn 7$$!3=aDX&o#)RB\"F/$!3`ZEUYWHu8Fgn7$$!3tnuihO%H7\"F/$!3&R@$*3qeJU\"Fgn 7$$!3FZPq5`lC5F/$!3]y@,n_Sp9Fgn7$$!3L\">[(=Tr2tB:Fgn7 $$!3/Q/[2BIT\")Fgn$!3#*Qjnt9Lz:Fgn7$$!3_\"\\\\$)Hp)pqFgn$!3\\aqxb=%=k \"Fgn7$$!3\"==!yI$ff3'Fgn$!3Xq0QJ)oPq\"Fgn7$$!3I`q84&zY-&Fgn$!3+$zT7_C gx\"Fgn7$$!3x)*)4Fgn7$$!3WK^&y8xi#>Fgn$!3t/>SeX*p-#Fgn7$$!3N/Lt\\ygb&)Fdil$!3_J!*4o'f 58#Fgn7$$\"3[\\opiFE>>Fdil$!3GRpi*)RvVAFgn7$$\"3%pL=]@WFgn$!3XVW:9^$H&GFgn7$$\"3+yN5ZMz*R&Fgn$!3'\\ W0U'p4WIFgn7$$\"3v7oB\"eC$pkFgn$!3z0))Ggtr%G$Fgn7$$\"3Wbe%z-qcZ(Fgn$!3 Gq\"*4u%['[NFgn7$$\"3]))o,&)4fF&)Fgn$!3ODx0K*GS(QFgn7$$\"32@%*>%*f.c&* Fgn$!39x\"e-!\\^bUFgn7$$\"3bs!\\2EsK1\"F/$!3K6rx?\"\\Yu%Fgn7$$\"3BpHLy *pp;\"F/$!3l7$*zM_FN`Fgn7$$\"3'Qrb>(e,t7F/$!3st&))*3r`8hFgn7$$\"3wYy._ O=y8F/$!3@?i\"\\X>v9(Fgn7$$\"3D8-j>>#[Z\"F/$!3%QO_*)*yti%)Fgn7$$\"3WqR \"*z-e&e\"F/$!3/I^tP+Xs5F/7$$\"3]g#Qmen*3>@F/7$$\"3[B5>>M\"3%=F/$!3\"\\S#H*zr>z#F/7$$\"3*oO 0N]g8*=F/$!3W#)ynKC+\"4%F/7$$\"3ED!HhP?&=>F/$!3xp$>)4)pYX&F/7$$\"3U$o_ ([-oX>F/$!36^cZ8W+#=)F/7$$\"3g7X1&=g#f>F/$!3$*R'H;)Q$44\"F_\\n7$$\"3dT jP@,%G(>F/$!3?9J%>q+kj\"F_\\n7$$\"30cA`*3I'z>F/$!39)RA>Wn==#F_\\n7$$\" 3wq\")od+U')>F/$!3sS%y3o+GF$F_\\n7$$\"3+GhwT]\")*)>F/$!3eux\\)fLPO%F_ \\n7$$\"3Y&3We-5K*>F/$!3!\\kF(p%)fXlF_\\n7$$\"3rU?#*4]g'*>F/$!3+o`8P&= \"48Fc^n7$$\"3&******R*******>F/F[`n7gn7$$\"30+++1+++?F/Fb`n7$$\"3w@O1 n6o+?F/$\"3Qy(=/[\\Z_'Fc^n7$$\"3-Vs7GBO,?F/$\"3-NRW3\"*QiKFc^n7$$\"3Gk 3>*[V?+#F/$\"3w;dDl#H\\<#Fc^n7$$\"3a&[a-lCF+#F/$\"3RMVIY\")>J;Fc^n7$$ \"31Gm&o'=FF_\\n7$$\"3%R)e.grz@?F/$\"34_kt 6;+R?F_\\n7$$\"3]?[i`*z7.#F/$\"3OxV*[Ig3U\"F_\\n7$$\"31dP@ZFwS?F/$\"3M @uzW,K!4\"F_\\n7$$\"3XAV+\"3uC@F /$\"3\"G\"*)p)QVHc$F/7$$\"3awMDQRDX@F/$\"3q%)3&*>bxfIF/7$$\"3taGzW'ok; #F/$\"3p)>TbyQ)pEF/7$$\"353$*\\:`h(=#F/$\"3q6Vq]M\"*oBF/7$$\"3#*HrdlsO 4AF/$\"3a!yU>:)zA@F/7$$\"3J3b/&RE&GAF/$\"3`45kF/7$$\"3qH%**p.&4] AF/$\"3\")\\'Gf@-rx\"F/7$$\"3V'[p=C_uETKe7F/7$$\"3?KD(=F6aP#F/ $\"3.^l&py')Q=\"F/7$$\"3')>H'>%*o]R#F/$\"3qwMhe%z\\7\"F/7$$\"3$3=D;_Om T#F/$\"3crkbVQum5F/7$$\"3Wc>vdRMT1&* \\#F/$\"3x[rIKCn(*))Fgn7$$\"3RVBmUQb@DF/$\"3^&ogk\"[a@&)Fgn7$$\"3wA0j% >Y2a#F/$\"3*G%>s%=&4>#)Fgn7$$\"3\"*>yagWZhDF/$\"3GN%)>iim:zFgn7$$\"3:n E](y))Ge#F/$\"3ZF#z,Oe[i(Fgn7$$\"3'*opiG&QQg#F/$\"3g\\8z7%>.O(Fgn7$$\" 3Q\\.Q![3Ti#F/$\"3m!o?H6p77(Fgn7$$\"3R6.(e![hYEF/$\"3]O]vgMStoFgn7$$\" 3p1BLSx$om#F/$\"3.J1*)oj&\\m'Fgn7$$\"3/?%p$R+V)o#F/$\"3h9%e`+8fX'Fgn7$ $\"3P\"pV:(e*zq#F/$\"3RR$['40]xiFgn7$$\"3&4oBT\\'QHFF/$\"3+3-(42,M4'Fg n7$$\"38`7'HS8&\\FF/$\"3GsfKRJxHfFgn7$$\"3:*o,?#=bqFF/$\"3\"y-M^@syw&F gn7$$\"3+4m3A27\"z#F/$\"3vZuwx#4zh&Fgn7$$\"3Bt!\\UXaE\"GF/$\"3#\\)R%z \"e/paFgn7$$\"3^-jm(*RRLGF/$\"3N6R')4^%HL&Fgn7$$\"3OZ!*GwJgaGF/$\"3i>7 QVVf+_Fgn7$$\"358X?KnjvGF/$\"3\\o/!Rarc2&Fgn7$$\"3&Q@IcQk\\*GF/$\"3Y>e vEt0m\\Fgn7$$\"3>P1edg6 k0JOu%Fgn7$$\"3[!ow$oK0eHF/$\"33))fTzo.RYFgn7$$\"3Hn`+-@FyHF/$\"3W<&z< rdJa%Fgn7$FgtFdclF^\\l-%+AXESLABELSG6$Q\"x6\"Q!6\"-%&TITLEG6#Q3singula r~solutions6\"-%%VIEWG6$;F(Fgt;$!\"\"F*$\"\"\"F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Cu rve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" }}}}{EXCHG {PARA 259 "" 0 "" {TEXT 276 123 "The hyperbolas are envelopes of the \+ non-singular solutions. The horizontal lines are limits of the nonsing ular solutions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 " \+ You can check with the closed form solution: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "general_sol := 'A'*tan('A'^3*x+'B');\nodetest(y(x)=ge neral_sol, ode2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,general_solG*& %\"AG\"\"\"-%$tanG6#,&*&)F&\"\"$F'%\"xGF'F'%\"BGF'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 81 "Travelling wave of the Kort eg de Vries equation - Weierstrass elliptic function." }}{PARA 0 "" 0 "" {TEXT -1 53 "We look for the equation of a solitary wave solution " }{XPPEDIT 18 0 "y(x-c*t)" "6#-%\"yG6#,&%\"xG\"\"\"*&%\"cGF(%\"tGF(! \"\"" }{TEXT -1 24 " of the KdV equation: " }{XPPEDIT 18 0 "u[t]-6*u *u[x]+u[x,x,x] = 0;" "6#/,(&%\"uG6#%\"tG\"\"\"*(\"\"'F)F&F)&F&6#%\"xGF )!\"\"&F&6%F.F.F.F)\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 7 "Let " }{XPPEDIT 18 0 "u(x,t) = 2*y(s)-c/6" "6#/-%\"uG6$%\"xG%\"t G,&*&\"\"#\"\"\"-%\"yG6#%\"sGF,F,*&%\"cGF,\"\"'!\"\"F4" }{TEXT -1 11 " where " }{XPPEDIT 18 0 "s = x-c*t;" "6#/%\"sG,&%\"xG\"\"\"*&%\"cG F'%\"tGF'!\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 69 "We proceed to show first how to obtain th e equation for y(s), namely " }}{PARA 0 "" 0 "" {MPLTEXT 0 21 12 " \+ " }{XPPEDIT 18 0 " Wp := diff(y(s),s)^2-4*y(s)^3+g[2]*y(s) +g [3]=0" "6#>%#WpG/,**$-%%diffG6$-%\"yG6#%\"sGF.\"\"#\"\"\"*&\"\"%F0*$-F ,6#F.\"\"$F0!\"\"*&&%\"gG6#F/F0-F,6#F.F0F0&F:6#F6F0\"\"!" }{TEXT -1 13 " where " }{XPPEDIT 18 0 "g[2]" "6#&%\"gG6#\"\"#" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "g[3]" "6#&%\"gG6#\"\"$" }{TEXT -1 26 " are arbitrary constants." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "When " }{XPPEDIT 18 0 "g[2]^3 <> 27*g[3]^2;" "6#0*$&% \"gG6#\"\"#\"\"$*&\"#F\"\"\"*$&F&6#F)F(F," }{TEXT -1 20 " , that is wh en " }{XPPEDIT 18 0 "p := -4*r^3+g[2]*r+g[3];" "6#>%\"pG,(*&\"\"% \"\"\"*$%\"rG\"\"$F(!\"\"*&&%\"gG6#\"\"#F(F*F(F(&F/6#F+F(" }{TEXT -1 217 " has only simple roots, the non-singular solutions are t ranslation of the Weirstrass elliptic function (WeierstrassP). We stud y the singular solutions in this generic case and in the degenerate ca ses, namely " }{XPPEDIT 18 0 "g[3]=g^3,g[2]=3*g^2" "6$/&%\"gG6#\"\"$*$ F%F'/&F%6#\"\"#*&F'\"\"\"*$F%F,F." }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g[3] = 0,g[2] = 0;" "6$/&%\"gG6#\"\"$\"\"!/&F%6#\"\"#F(" }{TEXT -1 43 " and see how their analyic properties vary." }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 27 "KdV -> Weierstrass Elliptic" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "kdv := diff(u(t,x),t)-6*u(t,x)*diff(u(t,x),x)+diff(u(t,x),`$`(x ,3)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$kdvG,(-%%diffG6$-%\"uG6$%\"tG%\"xGF,\"\"\"*(\"\"'F.F )F.-F'6$F)F-F.!\"\"-F'6$F)-%\"$G6$F-\"\"$F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "wave := 1/2*expand(convert(eval(eval(kdv, u(t,x)= \+ 2*y(x-c*t)-c/6), x=c*t+s),diff));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %%waveG,&*&-%%diffG6$-%\"yG6#%\"sGF-\"\"\"F*F.!#7-F(6$F*-%\"$G6$F-\"\" $F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "This equation can be integ rated into " }{XPPEDIT 18 0 " diff(y(s),s,s)-6*y(s)^2+g[2]/2 =0 " "6#/ ,(-%%diffG6%-%\"yG6#%\"sGF+F+\"\"\"*&\"\"'F,*$-F)6#F+\"\"#F,!\"\"*&&% \"gG6#F2F,F2F3F,\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 18 "which in turn ha s " }{XPPEDIT 18 0 "2*diff(y(s),s)" "6#*&\"\"#\"\"\"-%%diffG6$-%\"yG6# %\"sGF,F%" }{TEXT -1 36 " as an integrating factor. Finally, " } {XPPEDIT 18 0 "y(s)" "6#-%\"yG6#%\"sG" }{TEXT -1 11 " satisfies " } {XPPEDIT 18 0 " diff(y(s),s) ^2-4*y(s)^3+g[2]*y(s) +g[3]=0" "6#/,**$- %%diffG6$-%\"yG6#%\"sGF,\"\"#\"\"\"*&\"\"%F.*$-F*6#F,\"\"$F.!\"\"*&&% \"gG6#F-F.-F*6#F,F.F.&F86#F4F.\"\"!" }{TEXT -1 24 ". We verify this re sult." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Wp := diff(y(s),s)^2-4*y(s)^3+g[2]*y(s) +g[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#WpG,**$)-%%diffG6$-%\"yG6#%\"sGF.\"\"#\"\"\"F0*& \"\"%F0)F+\"\"$F0!\"\"*&&%\"gG6#F/F0F+F0F0&F86#F4F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "ode2 :=simplify(diff(Wp,s)/(2*(diff(y(s), s))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ode2G,(-%%diffG6$-%\"yG6# %\"sG-%\"$G6$F,\"\"#\"\"\"*&\"\"'F1)F)F0F1!\"\"*&#F1F0F1&%\"gG6#F0F1F1 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "wave - diff(%,s);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 15 "Generic case : " }{XPPEDIT 18 0 "g2^3<>27*g3^2" "6#0*$%#g 2G\"\"$*&\"#F\"\"\"*$%#g3G\"\"#F)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Wp := diff(w(z),z)^2-4*w(z)^3+g2*w(z) +g3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#WpG,**$)-%%diffG6$-%\"wG6#%\"zGF.\"\"#\" \"\"F0*&\"\"%F0)F+\"\"$F0!\"\"*&%#g2GF0F+F0F0%#g3GF0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "with(diffalg):\nK := field_extension(tra nscendental_elements=[g2,g3]):\nR := differential_ring(ranking=[w],der ivations=[z],notation=diff,field_of_constants=K):\nequations(Rosenfeld _Groebner([Wp],R));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7#,**$)-%%dif fG6$-%\"wG6#%\"zGF.\"\"#\"\"\"F0*&\"\"%F0)F+\"\"$F0!\"\"*&%#g2GF0F+F0F 0%#g3GF07#,(*$F3F0F2F6F5F8F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "equations(essential_components(Wp,R));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7#,**$)-%%diffG6$-%\"wG6#%\"zGF.\"\"#\"\"\"F0*&\"\"%F 0)F+\"\"$F0!\"\"*&%#g2GF0F+F0F0%#g3GF07#,(*$F3F0F2F6F5F8F5" }}}{PARA 14 "" 0 "" {TEXT -1 32 "There are three singular zeros: " }{TEXT 25 7 "w(z)=ri" }{TEXT -1 8 ", where " }{TEXT 25 2 "ri" }{TEXT -1 26 " is on e of the root of of " }{TEXT 25 13 "4*r^3-g2*r-g3" }{TEXT -1 104 ". Th ere are all three essential singular zeros. As such they are envelopes of the non-singular zeros of " }{TEXT 25 2 "Wp" }{TEXT -1 54 ". This \+ can be observed by plotting the real zeros for " }{TEXT 25 10 "g2=4,g3 =-1" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 466 "sing sol := \{plot(\{fsolve(-4*r^3+4*r-1,r)\},z=-3..3, color=red, thickness =2)\}:\ngalsol1 :=\{seq(plots[odeplot](dsolve(\{diff(w(z),z,z)-6*w(z)^ 2+2,w(ic)=-1,D(w)(ic)=-1.0\},\n w(z),type=numeric),-3..3,color=blu e),ic=0..4)\}:\ngalsol2 := \{seq(plots[odeplot](dsolve(\{diff(w(z),z, z)-6*w(z)^2+2,w(ic)=2,D(w)(ic)=-5.0\},\n w(z),type=numeric),ic-0.1 5..ic+1.7,color=green),ic=-3..1)\}:\nplots[display](singsol union gals ol1 union galsol2,title=\"g2=4,g3=-1\",view=[-3..3,-3/2..3]);" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "62-%'CURVESG6$7T7$ $!\"$\"\"!$!3?oFd[yzl)*!#=7$$!+?5bxG!\"*$!35r6[=*yI3\"!#<7$$!+S?5bFF1$ !321__F,>06F47$$!+gIlKEF1$!31IjXHeq[5F47$$!+!3/-^#F1$!3=2b@vV8U#*F-7$$ !++^v(Q#F1$!3DrNJ\"36<`(F-7$$!+?hIlAF1$!3R\"*f%=@J\"3cF-7$$!+Sr&G9#F1$ !32U7WVAt&p$F-7$$!+g\"3/-#F1$!3Bf8G\\%*Ga>F-7$$!+!=fz*=F1$!3'GB%e@8gOZ !#>7$$!++-^vF-7$$!+?Ur&G%F^q$!3jaZ* y5%\\*o$F-7$$!+CWAhIF^q$!3RMzn1Qa,cF-7$$!+GYtO=F^q$!3?'*pcWNVDvF-7$$!* K[C7'F^q$!3Se!)*)H;+P#*F-7$$\"*k\\C7'F^q$!3(H[Hj8!R[5F47$$\"+gZtO=F^q$ !3cJhK+(H^5\"F47$$\"+cXAhIF^q$!3([WRnO&G$3\"F47$$\"+_Vr&G%F^q$!3@i!3&* )>9q)*F-7$$\"+[T?5bF^q$!3XF8rr3#pL)F-7$$\"+WRpMnF^q$!3[p#fm(f!zZ'F-7$$ \"+SP=fzF^q$!3e_<:6O[OXF-7$$\"+ONn$=*F^q$!3g=2PDCg.FF-7$$\"+Lj\"3/\"F1 $!3dXe%*3zT*4\"F-7$$\"+8`Ej6F1$\"3O1]is)zY>#Fgn7$$\"+$H9dG\"F1$\"3A1`i u!z3C\"F-7$$\"+tK;39F1$\"3qfnh92!p(>F-7$$\"+`AhI:F1$\"3SjKbrQ(zW#F-7$$ \"+L71`;F1$\"3vw!\\0pnBn#F-7$$\"+8-^vfz*=F1 $\"3]r\\v!pP4T#F-7$$\"+t\"3/-#F1$\"3UxiLj&*R8>F-7$$\"+`r&G9#F1$\"3\\Gj 4+/w[6F-7$$\"+LhIlAF1$\"3%=;\"*48+no*!#?7$$\"+8^v(Q#F1$!3TBkM!o&o_7F-7 $$\"+$4/-^#F1$!35O?!oz.T)GF-7$$\"+tIlKEF1$!3G*\\NS\"4+NZF-7$$\"+`?5bFF 1$!3ViUC\"F4F^\\l7$$!3-+ +DhkaI6F4F^\\l7$$!3s******\\XF`**F-F^\\l7$$!3u*******>#z2))F-F^\\l7$$! 3R++]7RKvuF-F^\\l7$$!3t,+++P'eH'F-F^\\l7$$!3q)***\\7*3=+&F-F^\\l7$$!3[ )***\\PFcpPF-F^\\l7$$!3:)****\\7VQ[#F-F^\\l7$$!32)***\\i6:.8F-F^\\l7$$ !3Xb+++v`hHFcyF^\\l7$$\"3_****\\(QIKH\"F-F^\\l7$$\"38****\\7:xWCF-F^\\ l7$$\"3F,++vuY)o$F-F^\\l7$$\"3*y******4FL(\\F-F^\\l7$$\"3@)****\\d6.B' F-F^\\l7$$\"3s****\\(o3lW(F-F^\\l7$$\"37*****\\A))oz)F-F^\\l7$$\"3e*** ***Hk-,5F4F^\\l7$$\"36+++D-eI6F4F^\\l7$$\"3t***\\(=_(zC\"F4F^\\l7$$\"3 M+++b*=jP\"F4F^\\l7$$\"3f***\\(3/3(\\\"F4F^\\l7$$\"33++vB4JB;F4F^\\l7$ $\"3u*****\\KCnu\"F4F^\\l7$$\"3s***\\(=n#f(=F4F^\\l7$$\"3P+++!)RO+?F4F ^\\l7$$\"30++]_!>w7#F4F^\\l7$$\"3N++v)Q?QD#F4F^\\l7$$\"3G+++5jypBF4F^ \\l7$$\"3<++]Ujp-DF4F^\\l7$$\"3++++gEd@EF4F^\\l7$$\"39++v3'>$[FF4F^\\l 7$$\"38++D6EjpGF4F^\\l7$$\"\"$F*F^\\l-Fc[l6&Fe[lFg[lFf[lFf[l-%*THICKNE SSG6#\"\"#-F$6%7S7$F($\"3]+++`Vlv$)F-7$Fa\\lFjel7$Fd\\lFjel7$Fg\\lFjel 7$Fj\\lFjel7$F]]lFjel7$F`]lFjel7$Fc]lFjel7$Ff]lFjel7$Fi]lFjel7$F\\^lFj el7$F_^lFjel7$Fb^lFjel7$Fe^lFjel7$Fh^lFjel7$F[_lFjel7$F^_lFjel7$Fa_lFj el7$Fd_lFjel7$Fg_lFjel7$Fj_lFjel7$F]`lFjel7$F``lFjel7$Fc`lFjel7$Ff`lFj el7$Fi`lFjel7$F\\alFjel7$F_alFjel7$FbalFjel7$FealFjel7$FhalFjel7$F[blF jel7$F^blFjel7$FablFjel7$FdblFjel7$FgblFjel7$FjblFjel7$F]clFjel7$F`clF jel7$FcclFjel7$FfclFjel7$FiclFjel7$F\\dlFjel7$F_dlFjel7$FbdlFjel7$Fedl Fjel7$FhdlFjel7$F[elFjel7$F^elFjelF`elFbel-F$6%7S7$F($!3*******>()fr5 \"F47$Fa\\lF`il7$Fd\\lF`il7$Fg\\lF`il7$Fj\\lF`il7$F]]lF`il7$F`]lF`il7$ Fc]lF`il7$Ff]lF`il7$Fi]lF`il7$F\\^lF`il7$F_^lF`il7$Fb^lF`il7$Fe^lF`il7 $Fh^lF`il7$F[_lF`il7$F^_lF`il7$Fa_lF`il7$Fd_lF`il7$Fg_lF`il7$Fj_lF`il7 $F]`lF`il7$F``lF`il7$Fc`lF`il7$Ff`lF`il7$Fi`lF`il7$F\\alF`il7$F_alF`il 7$FbalF`il7$FealF`il7$FhalF`il7$F[blF`il7$F^blF`il7$FablF`il7$FdblF`il 7$FgblF`il7$FjblF`il7$F]clF`il7$F`clF`il7$FcclF`il7$FfclF`il7$FiclF`il 7$F\\dlF`il7$F_dlF`il7$FbdlF`il7$FedlF`il7$FhdlF`il7$F[elF`il7$F^elF`i lF`elFbel-F$6$7T7$F($\"31mx3ii3X?F-7$F/$\"3vm\"e!))30T8F-7$F6$\"3ZcMC7 tGQNFgn7$F;$!3R'3&*HP^NI*Fgn7$F@$!3$GVS%*oQJ]#F-7$FE$!3PE%4j*H=9VF-7$F J$!3'*))[Qz'p9D'F-7$FO$!3U<1Rn]'>8)F-7$FT$!3d)*QU`\"zjr*F-7$FY$!3\"[hL b]gc2\"F47$Fin$!30aMDr')o16F47$F^o$!3'*>E$zBL)e5F47$Fco$!3I'**eS0/6T*F -7$Fho$!3w<>='Qs2u(F-7$F]p$!3_kzHYi4HeF-7$Fbp$!3)e)Qj@z01RF-7$Fgp$!3/8 '4%*e=&R@F-7$F\\q$!3f#3cnusxE'Fgn7$Fbq$\"3]Oy``o=FfFgn7$Fgq$\"3.z\\%oW pp^\"F-7$F\\r$\"3&pMFu*p*>$Fgn7$Fdt$!3Sjv\"3l'4n,5F47$Faw$!3-(3Fe?0<`)F-7$Ffw$!3+pi45Ax'p'F-7$F[x$!3()4 %p`&p$Qv%F-7$F`x$!3lzPmz\\I,HF-7$Fex$!3A9)H#QOMn7F-7$Fjx$\"3fq[i+#e-^) Fcy7$F_y$\"3!GZ,/3'))R6F-7$Fey$\"3^2!))=#QC2>F-7$Fjy$\"3x&z)ehdH2CF-7$ F_z$\"3@`'3MjI*eEF-7$Fdz$\"3\"*3rAYWQtEF-7$Fiz$\"3'*)\\0X&yL^CF-7$F^[l $\"33zgP*)pv#)>F-Fb[l-F$6$7T7$F($!3:0qt&H,zo\"F-7$F/$!3R\\/w.&zEb#Fgn7 $F6$\"3$zq'38\"*z1))Fgn7$F;$\"3vM`\\R@#[s\"F-7$F@$\"34GjLXW!fH#F-7$FE$ \"3V5Z-Li'Rh#F-7$FJ$\"3G\\;cBSv#p#F-7$FO$\"3CG7Tl&ff`#F-7$FT$\"3h'\\gE 8[j8#F-7$FY$\"3!y8l]pUF-7$F`s$!3yDQQru7=CF-7$Fes$!3td: 'f$eD!f)Fgn7$Fjs$\"3Wu>#=BqB5%Fgn7$F_t$\"35Vbcw.&GQ\"F-7$Fdt$\"3g'ySE& *\\K2#F-7$Fit$\"3()3t#=n2@]#F-7$F^u$\"3Q/g*zgXmo#F-7$Fcu$\"3\\%psH+6`j #F-7$Fhu$\"3UbW518pXBF-7$F]v$\"3%Q^6*f030=F-7$Fbv$\"3LbLiQumQ**Fgn7$Fg v$!3_'p`_.**Q2\"Fgn7$F\\w$!3#4*[*eW7g]\"F-7$Faw$!3\")>%z5[3*zJF-7$Ffw$ !3xgOHs!Qp0&F-7$F[x$!3&zB_>x3x*pF-7$F`x$!3Nf]bz:&Rz)F-7$Fex$!3J\"z1\\X e&>5F47$Fjx$!3NfCZH'pp4\"F47$F_y$!3$\\EYL0ft4\"F47$Fey$!3()*Q3qU_1-\"F 47$Fjy$!3*\\&H3I'*)*4))F-7$F_z$!3r\"4,^\">P;qF-7$Fdz$!3x&fhO[0f2&F-7$F iz$!3\")>`ii(eu>$F-7$F^[l$!3B*=k6ZF6_\"F-Fb[l-F$6$7T7$F($\"3on&3OCE^2# F-7$F/$\"3'Q`*\\3Yk&Q\"F-7$F6$\"3:74!H%Q:STFgn7$F;$!3cYWZA9SU&)Fgn7$F@ $!3H;_J6\\T7CF-7$FE$!3\\B\"y8wXH@%F-7$FJ$!3tC#p#G^\\ZhF-7$FO$!3Z2npV3s O!)F-7$FT$!3l71;K4VV'*F-7$FY$!3sb\"R!*=I=2\"F47$Fin$!3DDrs%yQq5\"F47$F ^o$!3M?\\7>kHj5F47$Fco$!3q91A1%=')[*F-7$Fho$!3O`OFZeMQyF-7$F]p$!3sk))4 4*=L$fF-7$Fbp$!3S(Q(p?]+1SF-7$Fgp$!3&3+YE9\\!GAF-7$F\\q$!3.>X;&*pB.qFg n7$Fbq$\"3V[s#=218N&Fgn7$Fgq$\"3+gWz$pJ[Z\"F-7$F\\r$\"3:%)=!G)>dM@F-7$ Far$\"3Ejvj#)o-NDF-7$Ffr$\"3m\\(H57BEp#F-7$F[s$\"3!zR#e%)3j9EF-7$F`s$ \"3u<F_\"pB%Q$F-7$F^u$!3yJ-Q?!RoF &F-7$Fcu$!3WmPu'pUG@(F-7$Fhu$!3%pR(Q?ZM+'=\"e*z'F-7$F[x$!3/CM,qn\"o&[F-7$F`x$!3oN4Z;dd&* HF-7$Fex$!3&o/I#GY$yM\"F-7$Fjx$\"3YARJ8=IQ?Fcy7$F_y$\"3q_A*HRj44\"F-7$ Fey$\"3d%4\"4>\")>t=F-7$Fjy$\"3be?U'32qQ#F-7$F_z$\"3#3Oo@gV:l#F-7$Fdz$ \"3t1SdtEbyEF-7$Fiz$\"3!QooFl0$pCF-7$F^[l$\"3l`i*\\o-V,#F-Fb[l-F$6$7T7 $F($!3x#z(oeotL**F-7$F/$!3,+FkAIC'3\"F47$F6$!3+,k5,t7/6F47$F;$!3xdwEt, iV5F47$F@$!3d<,X;?Kg\"*F-7$FE$!3a?#F-7$F\\q$\"3UnE $=l'zl:F-7$Fbq$\"3><18LGh(f'Fgn7$Fgq$!3$fOll8C&3aFgn7$F\\r$!3z!HdO/Fd. #F-7$Far$!3KZ@W8cS)y$F-7$Ffr$!39ty=tbw0dF-7$F[s$!3ft^RG%\\Wi(F-7$F`s$! 3@\\!*H*z5F47$Fdt$!361<%eie1!)*F-7$Fit$!3L)zvB'3cV#)F-7$F^u$!3t`h-Q=HujF -7$Fcu$!3'R8!fqgWMWF-7$Fhu$!3s8`J/KP6EF-7$F]v$!3bk+\"F4Fb[l-F$6$7T7 $$!$:$!\"#$\"3V\\(f(f;;)3$F47$$!+)*[C7JF1$\"3\"*yLK51\"[t#F47$$!+'z*[u IF1$\"3M-([d#F47$$!+#fz*)*HF1$\"3\\_ ]M#R4\\*>F47$$!+!\\C7'HF1$\"3(*ounq'o:#=F47$$!+)QpM#HF1$\"3+MB5;)eQn\" F47$$!+'G9d)GF1$\"3[DP0o\"Qta\"F47$$!+%=fz%GF1$\"3bv4Y\\D]Q9F47$$!+#3/ -\"GF1$\"3k-AA9\"fXM\"F47$$!+!)*[Cx#F1$\"3[:%Q&y\"F47$$!+w(Qpp#F1$\"3*>h/+Ld;8\"F47$$!+uO=fEF1$\"3%)G([miW' y5F47$$!+s&G9i#F1$\"3b(\\*=kiuK5F47$$!+qMn$e#F1$\"3*ouF2U38$**F-7$$!+o $=fa#F1$\"3]Sy/n&*4\"f*F-7$$!+mK;3DF1$\"32W8#3+S6I*F-7$$!+k\"3/Z#F1$\" 3xHyi6Btc!*F-7$$!+iIlKCF1$\"3R)*f\"[[HS&))F-7$$!+gz*[R#F1$\"36wc!G'*=* *o)F-7$$!+eG9dBF1$\"3FQe=&fK>c)F-7$$!+cxQ>BF1$\"3^tzg&px\"o%)F-7$$!+aE j\"G#F1$\"3r\\&*[)o&G2%)F-7$$!+_v(QC#F1$\"3h]@$H\"ePy$)F-7$$!+]C71AF1$ \"3iF/HT@.\"Q)F-7$$!+[tOo@F1$\"3kM,FIGH:%)F-7$$!+YAhI@F1$\"3C;AB/6l\"[ )F-7$$!+Wr&G4#F1$\"3Y%H;j,o5e)F-7$$!+U?5b?F1$\"3?7$*=#e(*\\r)F-7$$!+Sp M5BJZDa)))F-7$$!+Q=fz>F1$\"3Heu_LJ#\\4*F-7$$!+On$=%>F1$\"311 2&e+?nM*F-7$$!+M;3/>F1$\"35%3#pqvzW'*F-7$$!+KlKm=F1$\"3SNu!HB5S***F-7$ $!+I9dG=F1$\"3z:>5$)R-S5F47$$!+Gj\"3z\"F1$\"3'ox#\\*Gfq3\"F47$$!+E71`< F1$\"3b(fQ\")Hl89\"F47$$!+ChI:6'*ppXf8F47$$!+=3/-;F1$\"3&[')4;rSdX\"F47$$! +;dGk:F1$\"3m!*4qR`Mn:F47$$!+91`E:F1$\"3[Eoz!pqrp\"F47$$!+7bx)[\"F1$\" 3ouK)*H6')[=F47$$!+5/-^9F1$\"3Gno*o\\Wq-#F47$$!+3`E89F1$\"3,f>+.jmPAF4 7$$!+1-^v8F1$\"3.\"z0a3l%)[#F47$$!+/^vP8F1$\"3QAtw18m*y#F47$$!+-+++8F1 $\"3S*eU%4o(\\:$F4-Fc[l6&Fe[lFf[lFg[lFf[l-F$6$7T7$$!$:#F\\coF]co7$$!+) *[C7@F1Fbco7$$!+'z*[u?F1Fgco7$$!+%pMn.#F1F\\do7$$!+#fz*)*>F1Fado7$$!+! \\C7'>F1$\"3Ununq'o:#=F47$$!+)QpM#>F1$\"3*GL-h\")eQn\"F47$$!+'G9d)=F1F `eo7$$!+%=fz%=F1$\"3+u4Y\\D]Q9F47$$!+#3/-\"=F1$\"3J,AA9\"fXM\"F47$$!+! )*[Cx\"F1$\"3P9%Q&y\"F47$$!+w(Qpp\"F 1$\"3W5Y+ItlJ6F47$$!+uO=f;F1$\"3HF([miW'y5F47$$!+s&G9i\"F1$\"3A'\\*=ki uK5F47$$!+qMn$e\"F1$\"3dLxs?%38$**F-7$$!+o$=fa\"F1$\"3'\\#y/n&*4\"f*F- 7$$!+mK;3:F1$\"3_G8#3+S6I*F-7$$!+k\"3/Z\"F1$\"3A9yi6Btc!*F-7$$!+iIlK9F 1$\"3t\")f\"[[HS&))F-7$$!+gz*[R\"F1$\"3Mec!G'*=**o)F-7$$!+eG9d8F1$\"31 ;e=&fK>c)F-7$$!+cxQ>8F1$\"3(z%zg&px\"o%)F-7$$!+aEj\"G\"F1$\"3&3_*[)o&G 2%)F-7$$!+_v(QC\"F1$\"3U=@$H\"ePy$)F-7$$!+]C717F1$\"34#R!HT@.\"Q)F-7$$ !+[tOo6F1$\"3n%4q-$GH:%)F-7$$!+YAhI6F1$\"3$=pqvzW'*F-7$$!+?`Ej')F^q$\"3[Nt!HB5S***F-7$$!++Vr&G) F^q$\"3Z/>5$)R-S5F47$$!+!Gj\"3zF^q$\"3)Rw#\\*Gfq3\"F47$$!+gAhIvF^q$\"3 !HeQ\")Hl89\"F47$$!+S71`rF^q$\"3')zw<&[!)R?\"F47$$!+?-^vnF^q$\"3.bxtZ` ;w7F47$$!++#fzR'F^q$\"3m(4h*ppXf8F47$$!+!=3/-'F^q$\"3JR)4;rSdX\"F47$$! +gr&Gk&F^q$\"3,i4qR`Mn:F47$$!+ShIl_F^q$\"3G#z'z!pqrp\"F47$$!+?^v()[F^q $\"3PNK)*H6')[=F47$$!++T?5XF^q$\"3l?o*o\\Wq-#F47$$!+!3`E8%F^q$\"3R/>+. jmPAF47$$!+g?5bPF^q$\"3'[s0a3l%)[#F47$$!+S5bxLF^q$\"36Xsw18m*y#F47$$!+ ?+++IF^q$\"3P%\\U%4o(\\:$F4Fdbp-F$6$7T7$$!$:\"F\\co$\"3LU(f(f;;)3$F47$ $!+)*[C76F1$\"3etLK51\"[t#F47$$!+'z*[u5F1$\"3*=q[d#F47$$!+?fz*)**F^qFado7$$!++\\C7'*F^q$\"3'yYx1no:#=F47$ $!+!)QpM#*F^q$\"3yLB5;)eQn\"F47$$!+gG9d))F^q$\"3$fs`!o\"Qta\"F47$$!+S= fz%)F^q$\"3nu4Y\\D]Q9F47$$!+?3/-\")F^q$\"3w,AA9\"fXM\"F47$$!++)*[CxF^q $\"3/:%Q&yc)F-7$$!+gv(Q>$F^q$\"3Hrzg& px\"o%)F-7$$!+SlK;GF^q$\"3\\Z&*[)o&G2%)F-7$$!+?bxQCF^q$\"3R[@$H\"ePy$) F-7$$!++XAh?F^q$\"3SD/HT@.\"Q)F-7$$!+![tOo\"F^q$\"3UK,FIGH:%)F-7$$!+gC 718F^q$\"38:AB/6l\"[)F-7$$!+'R9dG*!#6$\"3F+'Rk,o5e)F-7$$!+#>/-^&Fcjq$ \"3!\\'4^#e(*\\r)F-7$$!+))RpMpqrp\"F47$$\"+$)[C7^F^q$\"3tcpGJ6')[=F 47$$\"+.fz*[&F^q$\"3)emK%)\\Wq-#F47$$\"+BpMneF^q$\"3F&\\B[ImwB#F47$$\" +Vz*[C'F^q$\"3>@Le(3l%)[#F47$$\"+j*[Ci'F^q$\"3)4M%R48m*y#F47$$\"+$)*** ***pF^q$\"3lrck7o(\\:$F4Fdbp-F$6$7T7$$!#:F\\co$\"3*\\uf(f;;)3$F47$$!+! )*[C7\"F^q$\"3\"\\PB.h5[t#F47$$!+'fz*[uFcjq$\"3nUrYv@'HW#F47$$!+#RpMn$ Fcjq$\"3N:]p%ev&*>#F47$$\"*73/-\"Fcjq$\"3'*Qxu\"R4\\*>F47$$\"+;5bxQFcj q$\"3)\\r++no:#=F47$$\"+?71`wFcjq$\"3Za%z`\")eQn\"F47$$\"+Ur&G9\"F^q$ \"3r&4Lu;Qta\"F47$$\"+i\"3/_\"F^q$\"3l$3E*[D]Q9F47$$\"+#=fz*=F^q$\"3&p #)fP6fXM\"F47$$\"+--^vAF^q$\"3*p(y8y\"F47$$\"+UAhIIF^q$\"3RaKqHtlJ6F47$$\"+iK;3MF^q$\"3*>b(QEYky5F47$$\"+ #G9dy$F^q$\"3(omjREYF.\"F47$$\"+-`EjTF^q$\"3G*)Gy=%38$**F-7$$\"+Aj\"3a %F^q$\"36$H$Ql&*4\"f*F-7$$\"+UtO=\\F^q$\"3-J(4%***R6I*F-7$$\"+i$=fH&F^ q$\"3_LmW5Btc!*F-7$$\"+#QpMn&F^q$\"3%p(o%Q[HS&))F-7$$\"+-/-^gF^q$\"3n: R.i*=**o)F-7$$\"+A9dGkF^q$\"3;R**f%fK>c)F-7$$\"+UC71oF^q$\"30P$*>&px\" o%)F-7$$\"+iMn$=(F^q$\"3(GC_#)o&G2%)F-7$$\"+#[C7c(F^q$\"3!RviG\"ePy$)F -7$$\"+-bxQzF^q$\"31WzQT@.\"Q)F-7$$\"+AlK;$)F^q$\"3Llf`IGH:%)F-7$$\"+U v(Qp)F^q$\"3,(>qY5^;[)F-7$$\"+i&G92*F^q$\"3J\"zKp,o5e)F-7$$\"+#ez*[%*F ^q$\"3)yR$*He(*\\r)F-7$$\"+-1`E)*F^q$\"3'zyETZDa)))F-7$$\"+i\"3/-\"F1$ \"3,&RFN8B\\4*F-7$$\"+kK;e5F1$\"3+N1&e+?nM*F-7$$\"+m$=f4\"F1$\"30.?pqv zW'*F-7$$\"+oMnL6F1$\"3OWt!HB5S***F-7$$\"+q&G9<\"F1$\"3e0>5$)R-S5F47$$ \"+sO=47F1$\"3JlF\\*Gfq3\"F47$$\"+u(QpC\"F1$\"3X%eQ\")Hl89\"F47$$\"+wQ p%G\"F1$\"3k\"ox^[!)R?\"F47$$\"+y*[CK\"F1$\"3.dxtZ`;w7F47$$\"+!3/-O\"F 1$\"3))*4h*ppXf8F47$$\"+#=fzR\"F1$\"3kU)4;rSdX\"F47$$\"+%G9dV\"F1$\"3! \\'4qR`Mn:F47$$\"+'QpMZ\"F1$\"3]'z'z!pqrp\"F47$$\"+)[C7^\"F1$\"3fRK)*H 6')[=F47$$\"+!fz*[:F1$\"3)f#o*o\\Wq-#F47$$\"+#pMne\"F1$\"3s4>+.jmPAF47 $$\"+%z*[C;F1$\"3'Gt0a3l%)[#F47$$\"+'*[Ci;F1$\"3m_sw18m*y#F47$$\"+)*** ***p\"F1$\"3p.DW4o(\\:$F4Fdbp-F$6$7T7$$\"#&)F\\co$\"3WX(f(f;;)3$F47$$ \"+?5bx))F^q$\"3DwLK51\"[t#F47$$\"+S?5b#*F^qFgbq7$$\"+gIlK'*F^q$\"3dPn ;&ev&*>#F47$$\"+3/-,5F1Fado7$$\"+5bxQ5F1F[dp7$$\"+71`w5F1F`dp7$$\"+9dG 96F1F`eo7$$\"+;3/_6F1Fhdp7$$\"+=fz*=\"F1F]ep7$$\"+?5bF7F1Fbep7$$\"+AhI l7F1$\"3aSF%o$)*y#>\"F47$$\"+C71.8F1$\"3m5Y+ItlJ6F47$$\"+Ej\"3M\"F1$\" 3^F([miW'y5F47$$\"+G9dy8F1$\"3W'\\*=kiuK5F47$$\"+IlK;9F1$\"3zNxs?%38$* *F-7$$\"+K;3a9F1$\"32Ey/n&*4\"f*F-7$$\"+Mn$=\\\"F1$\"3uI8#3+S6I*F-7$$ \"+O=fH:F1$\"3X;yi6Btc!*F-7$$\"+QpMn:F1$\"3&R)f\"[[HS&))F-7$$\"+S?50;F 1$\"3nhc!G'*=**o)F-7$$\"+Ur&Gk\"F1$\"3R>e=&fK>c)F-7$$\"+WAh!o\"F1$\"3J ^zg&px\"o%)F-7$$\"+YtO=F1$\"30]iJ;!o5e)F-7$$\"+ez*[%>F 1$\"3Nj#*=#e(*\\r)F-7$$\"+gIl#)>F1$\"3UYI7taU&)))F-7$$\"+i\"3/-#F1$\"3 7'RFN8B\\4*F-7$$\"+kK;e?F1$\"36O1&e+?nM*F-7$$\"+m$=f4#F1$\"3%>+#pqvzW' *F-7$$\"+oMnL@F1F_\\s7$$\"+q&G9<#F1Fd\\s7$$\"+sO=4AF1$\"3`lF\\*Gfq3\"F 47$$\"+u(QpC#F1$\"3*[eQ\")Hl89\"F47$$\"+wQp%G#F1$\"3>\"ox^[!)R?\"F47$$ \"+y*[CK#F1Fh]s7$$\"+!3/-O#F1$\"3m*4h*ppXf8F47$$\"+#=fzR#F1$\"3UU)4;rS dX\"F47$$\"+%G9dV#F1$\"3,k4qR`Mn:F47$$\"+'QpMZ#F1$\"3R&z'z!pqrp\"F47$$ \"+)[C7^#F1$\"3:RK)*H6')[=F47$$\"+!fz*[DF1$\"3`Do*o\\Wq-#F47$$\"+#pMne #F1$\"3]2>+.jmPAF47$$\"+%z*[CEF1$\"3>IdS&3l%)[#F47$$\"+'*[CiEF1$\"3L^s w18m*y#F47$$\"+)******p#F1$\"3e+DW4o(\\:$F4Fdbp-%&TITLEG6#Q+g2=4,g3=-1 6\"-%+AXESLABELSG6$Q\"z6\"Q!6\"-%%VIEWG6$;F(F^el;$!0++++++]\"!#9F^el" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Cur ve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" }}}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {XPPEDIT 18 0 "g2^3=27*g3^2" "6#/*$%#g2G\"\"$*&\"#F \"\"\"*$%#g3G\"\"#F)" }}{PARA 14 "" 0 "" {TEXT -1 34 "We parameterise \+ the equation with " }{TEXT 25 1 "g" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 32 "Wps := subs(g2=3*g^2,g3=g^3,Wp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$WpsG,**$)-%%diffG6$-%\"wG6#%\"zGF.\"\"#\" \"\"F0*&\"\"%F0)F+\"\"$F0!\"\"*(F4F0)%\"gGF/F0F+F0F0*$)F8F4F0F0" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "Ks := field_extension(trans cendental_elements=[g]):\nRs := differential_ring(ranking=[w],derivati ons=[z],notation=diff,field_of_constants=Ks):\nequations(Rosenfeld_Gro ebner([Wps],Rs));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7#,**$)-%%diffG 6$-%\"wG6#%\"zGF.\"\"#\"\"\"F0*&\"\"%F0)F+\"\"$F0!\"\"*(F4F0)%\"gGF/F0 F+F0F0*$)F8F4F0F07#,(*$)F+F/F0F/*&F+F0F8F0F5*$F7F0F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "equations(essential_components(Wps,Rs)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7#,**$)-%%diffG6$-%\"wG6#%\"zGF. \"\"#\"\"\"F0*&\"\"%F0)F+\"\"$F0!\"\"*(F4F0)%\"gGF/F0F+F0F0*$)F8F4F0F0 7#,&F+F0F8F5" }}}{PARA 14 "" 0 "" {TEXT -1 30 "There are two singular \+ zeros: " }{TEXT 25 6 "w(z)=g" }{TEXT -1 5 " and " }{TEXT 25 9 "w(z)=-g /2" }{TEXT -1 2 ". " }{TEXT 25 6 "w(z)=g" }{TEXT -1 34 " is an essenti al singular zero of " }{TEXT 25 3 "Wps" }{TEXT -1 59 " and therefore i s an envelope of the non singular zeros of " }{TEXT 25 3 "Wps" }{TEXT -1 37 ". On the contrary, the singular zero " }{TEXT 25 10 "w(z) =-g/2 " }{TEXT -1 4 " of " }{TEXT 25 3 "Wps" }{TEXT -1 80 " is not essential and therefore is a limiting case of the non singular zeros of " } {TEXT 25 3 "Wps" }{TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT -1 33 "We ca n observe the situation for " }{TEXT 25 4 "g=-1" }{TEXT -1 1 ":" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 296 "singsol:=plot(\{-1,1/2\},co lor=red,thickness=2):\ngalsol1:=plot(\{seq(-1+3/2*tanh(1/2*(C-z)*sqrt( 6))^2,C=-1..1)\},z=-3..3,color=green):\ngalsol2:=plot(\{seq(-1+3/2*cot h(1/2*(C-z)*sqrt(6))^2,C=-1..1)\},z=-3..3,color=blue):\nplots[display] (\{singsol,galsol1,galsol2\},view=[-3..3,-3/2..3],title=\"g2=3,g3=-1\" );" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6--%'CURVESG 6$7do7$$!\"$\"\"!$\"3xRW'oZmm*\\!#=7$$!3*)*****\\2<#pG!#<$\"30mr#=*yS& *\\F-7$$!3$)***\\7bBav#F1$\"3zmn6=?$R*\\F-7$$!36++]K3XFEF1$\"3!3A,\"R' )p\"*\\F-7$$!3%)****\\F)H')\\#F1$\"3l98/U*>'))\\F-7$$!3#****\\i3@/P#F1 $\"3LHHW>MU%)\\F-7$$!3:++Dr^b^AF1$\"3c`2<3D;z\\F-7$$!3$****\\7Sw%G@F1$ \"39&*)>([v$=(\\F-7$$!3*****\\7;)=,?F1$\"3t#[C0)pah\\F-7$$!3.++DO\"3V( =F1$\"3OXLE0_bZ\\F-7$$!3#******\\V'zVyp%y#\\F-7$$!3)*****\\ d;%)G;F1$\"32'R0O,cW!\\F-7$$!3*)*****\\!)H%*\\\"F1$\"3))er\\>Z(*o[F-7$ $!3/+++vl[p8F1$\"3AuXWPS;?[F-7$$!3!******\\>iUC\"F1$\"3W;3vGG8cZF-7$$! 3-++DhkaI6F1$\"3=wF>#z2))F-$\"3'e$yNqs%GT%F-7$$!3R++]7RKvuF-$\"3d+PB?zS#>%F-7$$!3 t,+++P'eH'F-$\"3nBSPt!z;$RF-7$$!3q)***\\7*3=+&F-$\"3zb\"*Hcu'Gb$F-7$$! 3[)***\\PFcpPF-$\"3K7W$*etkwIF-7$$!3:)****\\7VQ[#F-$\"3Y\"opw&H>GCF-7$ $!32)***\\i6:.8F-$\"3#3Bs9?snm\"F-7$$!3Xb+++v`hH!#?$\"3f@o>ZL\\rj!#>7$ $\"3_****\\(QIKH\"F-$!3/\\&RVyZ$QoFbs7$$\"3L******\\4+p=F-$!3vE\"=zb8' R8F-7$$\"38****\\7:xWCF-$!3GrR:WhPT?F-7$$\"3?++v$\\>m1$F-$!3H34u.6UXGF -7$$\"3F,++vuY)o$F-$!3wmyS=#z$)o$F-7$$\"3e****\\(G(*3L%F-$!3SJmHO]['e% F-7$$\"3*y******4FL(\\F-$!3mV!eq,^S\\&F-7$$\"30)***\\P$>=g&F-$!3vr0L_R QpjF-7$$\"3@)****\\d6.B'F-$!3Q^/ji#>w?(F-7$$\"3(*)**\\785%QoF-$!38ut:l :2ezF-7$$\"3s****\\(o3lW(F-$!3FlRUYAaB')F-7$$\"3U***\\iX)p@\")F-$!3bx) GR]%QL#*F-7$$\"37*****\\A))oz)F-$!3u#fyNHu*y'*F-7$$\"3G)**\\PCK-5*F-$! 3.)RhTm3$>)*F-7$$\"3X(***\\iid.%*F-$!31V(*QklC?**F-7$$\"3['*\\(=F[_b*F -$!3;-jKk>eb**F-7$$\"3i'**\\7G?pq*F-$!3!)y$470!p!)**F-7$$\"3w'*\\i!H#f e)*F-$!3!e')Qzw,b***F-7$$\"3e******Hk-,5F1$!3B9x&\\Hw*****F-7$$\"3`** \\Pa1A<5F1$!33P=7#eHL***F-7$$\"3r***\\(y[TL5F1$!3)z(QDcb!\\(**F-7$$\"3 *)**\\7.\"4'\\5F1$!3E(*y8u?wW**F-7$$\"3%)****\\FL!e1\"F1$!3?Y\"))\\c$* H!**F-7$$\"3(****\\ix\">)4\"F1$!3=$y$zY!Q^y*F-7$$\"36+++D-eI6F1$!3G:(3 sE'zA'*F-7$$\"3#***\\(=sx#*=\"F1$!3qk6'3>P>A*F-7$$\"3t***\\(=_(zC\"F1$ !3[FxeR!)o(o\"3rsbF-7$$\"3%)***\\im&>g:F1$!3yu)GpUc7o%F-7$$\"33++vB4J B;F1$!3Ys^Tt&[oz$F-7$$\"3\"***\\PCw,&o\"F1$!3GRvFdp>cHF-7$$\"3u*****\\ KCnu\"F1$!3#)e))3EcR_@F-7$$\"3t**\\(=_D8\"=F1$!3AI6k'=s0O\"F-7$$\"3s** *\\(=n#f(=F1$!3VJt7K#pVE'Fbs7$$\"30+]P\\`9Q>F1$\"3!RRt\"z[>gAF_s7$$\"3 P+++!)RO+?F1$\"36\\s#>/Js8'Fbs7$$\"30++]_!>w7#F1$\"3z[:(eM5tk\"F-7$$\" 3N++v)Q?QD#F1$\"3U,v.fb>fCF-7$$\"3G+++5jypBF1$\"3RI)*oe>8c$F-7$$\"3++++gEd@EF1$\"3&>Sl1=%G6RF-7$$\"39++v3' >$[FF1$\"3p5PM/8#R>%F-7$$\"38++D6EjpGF1$\"3$RHV$)4DpR%F-7$$\"\"$F*$\"3 r];B*yU$fXF--%'COLOURG6&%$RGBG$F*F*$\"*++++\"!\")Fbbl-F$6$7do7$F(F\\bl 7$F/$\"3&Q_],VBjR%F-7$F5$\"3kFHk#>au?%F-7$F:$\"3/YRw<.GERF-7$F?$\"3*>x !4WBjZNF-7$FD$\"3?dSf$R+x/$F-7$FI$\"3e<)=@uF1$\"3)*>NfP[Ak=F_s7$FX$!3J` 8.(\\-4W'Fbs7$$!3)***\\i&G_!4=F1$!3GjJLWq[(Q\"F-7$Fgn$!3(=5c'40b*=#F-7 $$!3&****\\i/>jo\"F1$!3yCCpQ\\#)QHF-7$F\\o$!3(**>x^84.s$F-7$$!3%****\\ 7tNTc\"F1$!3hDyeNoUkF-7$Ffo$!3s\")4`0Jp.tF-7$$!3(******\\QuoI\"F1$!3`&Ghq^'zl!)F- 7$F[p$!3)[M=oIuPt)F-7$$!3'***\\7GVS(=\"F1$!3()y,V9vtO#*F-7$F`p$!3Se_AA !))Hi*F-7$$!32+vo4D>C&**F-7$$!3$**\\il?K\"H5F1$!3H2 F\\r2#4)**F-7$$!31]7yI)HA,\"F1$!3(3Beh?Nm***F-7$Fep$!3x(H(\\v(3&****F- 7$$!3s***\\iD*35)*F-$!3z4kVb!))=***F-7$$!3t****\\iR!pm*F-$!3h;*oVDj](* *F-7$$!3t***\\(o'=P_*F-$!3t&)H/zc2\\**F-7$$!3t*****\\PL0Q*F-$!3CC\"4^* *))R\"**F-7$$!3t****\\(yiT4*F-$!3kV()>hE)o\")*F-7$Fjp$!3b]2=.ao%o*F-7$ $!31++Dc!e:9)F-$!3NgB<\"4m*[#*F-7$F_q$!3bi!oLsZDl)F-7$$!31,+D1Qf&)oF-$ !3Y=(=a(*pI,)F-7$Fdq$!3Uu38^-%=H(F-7$$!3A++D1j$)[cF-$!3u(ps*[vpLkF-7$F iq$!3iS?p&ykT`&F-7$$!3f)****\\#eo&Q%F-$!3$))>7.z2Qm%F-7$F^r$!3rclC08b+ QF-7$$!3K)**\\7$HqEJF-$!3yHov4cMDHF-7$Fcr$!3HT*G3Se04#F-7$$!36)**\\P9( \\$*=F-$!3\"=j$eB%f&o8F-7$Fhr$!3Ni%G%4?FZpFbs7$F]s$\"3Ti!)3q0CF-7$Fht$\"3aX'[2,n1/$F-7$Fbu$\"3 )HIW\")eSKa$F-7$F\\v$\"3)QGXFqB]\"RF-7$Ffv$\"3>6LTlZ%o=%F-7$F`w$\"3.) \\J/138T%F-7$F^y$\"35%)\\PxHVgXF-7$F\\[l$\"39\"4)H2/nyYF-7$Ff[l$\"3pB[ *)))GKeZF-7$F`\\l$\"3'=JC+0JJ#[F-7$Fj\\l$\"3\"y78v%=Ao[F-7$Fd]l$\"3))* eZ)Rq:.\\F-7$F^^l$\"3BK/%ROh$G\\F-7$Fh^l$\"3SU#4*fBwZ\\F-7$Fb_l$\"3?N- 7s$p9'\\F-7$Fg_l$\"3JBAE*Ry<(\\F-7$F\\`l$\"3:c#*zFxFz\\F-7$Fa`l$\"3eG2 `$>*R%)\\F-7$Ff`l$\"3P+#\\CpK())\\F-7$F[al$\"3[z9\"4Gy:*\\F-7$F`al$\"3 Cq![o_DQ*\\F-7$Feal$\"3Yp-#Gc7a*\\F-7$FjalF+F^bl-F$6$7eo7$F($\"3/)GS%[ ]Vh\\F-7$F/$\"39zbIj#)*o%\\F-7$F5$\"3yCWY.#o)H\\F-7$F:$\"3`v&y<+JT!\\F -7$F?$\"3#>j[g%)=(o[F-7$FD$\"3.s/5eFd?[F-7$FI$\"38!H.`)fTgZF-7$FN$\"3e x6C(>Gqn%F-7$FS$\"3x(yhB^/1c%F-7$FX$\"3vW]S+Yl.WF-7$Fgn$\"3z=;GE!)=&=% F-7$F\\o$\"3f*G5.P*zHRF-7$Fao$\"3py&GGkM.b$F-7$Ffo$\"3r#\\JCFGN/$F-7$F [p$\"3bZ$*ye$eWS#F-7$F`p$\"3vzF*QnP%o;F-7$Fep$\"3o)\\(R1x;!o&Fbs7$Ffjl $\"3KG#y-8*[o@F_s7$Fjp$!3#[-Iv%HWPdFbs7$Fc[m$!3)\\0qmGirK\"F-7$F_q$!3F jeT(of@9#F-7$F[\\m$!3i#QYV')e*3HF-7$Fdq$!3()y*z]_=+r$F-7$Fc\\m$!3Dmd`j #z]h%F-7$Fiq$!3H1Ze;42HbF-7$F[]m$!3;_^J$*f\\'Q'F-7$F^r$!3i;))4?9y2sF-7 $Fc]m$!3`3&GPf'y)*zF-7$Fcr$!3]HY)=CCKp)F-7$F[^m$!3V@=!R!fO@#*F-7$Fhr$! 3$GX)of1IC'*F-7$$!3$p)*\\i:sw%)*Fbs$!3'ps?#R7!Ry*F-7$$!38$***\\(oKQm'F bs$!3Z)=h:-F0!**F-7$$!3C'*\\7`H\">2&Fbs$!3(3?d#*))oA%**F-7$$!3M***\\(= K**zMFbs$!3%=P.4n%ys**F-7$$!3W-]P%[t!))=Fbs$!3Q%p(35?)>***F-7$F]s$!3Q< E)3m-)****F-7$$\"3b%*\\7`MSd8Fbs$!3ZKcWR]&e***F-7$$\"3l%**\\ilg4,$Fbs$ !3dX8SR-iz**F-7$$\"3v%*\\Pfy^kYFbs$!3&4-7EV^6&**F-7$$\"3%[***\\i]2=jFb s$!3E!*)*)4lT0\"**F-7$$\"3/&**\\(o%*=D'*Fbs$!3&=X[aXmMz*F-7$Fds$!30<)3 F,0*H'*F-7$Fis$!3O\">UTU*pS#*F-7$F^t$!3T'z[')fs;t)F-7$Fct$!3\"3=!y/uBo !)F-7$Fht$!3(yIMc4l=J(F-7$F]u$!3d?yN`JNhkF-7$Fbu$!3W\\B'[*3;pbF-7$Fgu$ !3gA(*3W.X\"o%F-7$F\\v$!3-nB%)*[D2!QF-7$Fav$!3OP%G'or\">(HF-7$Ffv$!3[J 5!4)ppy@F-7$F[w$!3-kjcdPf]8F-7$F`w$!30BYmvVabeFbs7$Fjw$\"3))p&)o\"H)Gh WF_s7$F^y$\"3`t)pn9pq>'Fbs7$F\\[l$\"3_m:N#o!oo;F-7$Ff[l$\"3=O&fa%F-7$Fh^l$\"3Q]fSaq'fS%F-7$Fb_l$\"3\\>Bu#oH(fXF- 7$Fg_l$\"3KY3*oecjn%F-7$F\\`l$\"3,K+<&4JuuWtF17$F[p$\"3O/#>gSM pn\"!#;7$Fcgl$\"3MVI1c=(y%GF[en7$F`p$\"3,h-(3:!)z'eF[en7$$!3;](oa3WO6 \"F1$\"3:kML&zSJu(F[en7$F[hl$\"3QDYn!)*3&o5!#:7$$!3)*\\i!RLR)z5F1$\"3F 7\"pM_/)o:Fjen7$F`hl$\"3H)*>\"eC#eCDFjen7$$!3%[PM-x&[a5F1$\"3]y@9!>$[o LFjen7$Fehl$\"3%[H_kL.)=ZFjen7$$!3(\\7`WR$eP5F1$\"3H))e1..fzqFjen7$Fjh l$\"3*Qj\"HnHHy6!#97$$!3)[(=n=5o?5F1$\"3Al%posi!QBFcgn7$F_il$\"30+Y()H \"))eo'Fcgn7$$!3lPf$oB/!35F1$\"3Jw$\\2IY3c\"!#87$$!3-D1*GkyP+\"F1$\"3z VjYTBr.qFahn7$$!3%\\7`%*[Ib***F-$\"3!pWgYwve+&!#67$Fep$\"306WG2/H!e%Fa hn7$$!3W*\\ilAyu\"**F-$\"3\"[;pphi%o9Fahn7$$!3<**\\7.>o\"))*F-$\"3AChE 8'zK9(Fcgn7$$!3*))\\(ozb)e%)*F-$\"3)zku,&RI5UFcgn7$Fgil$\"3T%*G4%f(osF Fcgn7$$!3G+]P4m\\Q(*F-$\"3td!3Q(*HBY\"Fcgn7$F\\jl$\"3/=t,N$HG,*Fjen7$F ajl$\"3v()z*[1@$3WFjen7$Ffjl$\"3_LsYBV%fg#Fjen7$$!3t***\\73[tB*F-$\"30 qM-.8H>%)>Y/[A3tF17$Fiq$\"3JQMT$G^#QSF17$F^r$\"3*)R\"R5)\\NHE F17$Fcr$\"3e=()y^9qW=F17$Fhr$\"3A;;2?R)zT\"F17$F]s$\"3*o!okcM$f7\"F17$ Fds$\"3v&3%R#\\mtH*F-7$F^t$\"3it:&\\\"H#o8)F-7$Fht$\"31**)o(z'=PD(F-7$ Fbu$\"3Ca95mOX8mF-7$F\\v$\"3/))Hm1OdphF-7$Ffv$\"3PQ=H.LwfeF-7$F`w$\"3 \\3L-(fRFh&F-7$F^y$\"3-^X@vr$GX&F-7$F\\[l$\"3CR&eHzj$G`F-7$Ff[l$\"3#[q w0tMcC&F-7$F`\\l$\"3D`W`U$z*y^F-7$Fj\\l$\"3#zq@?6YH8&F-7$Fd]l$\"3LE/Ig AZ(4&F-7$F^^l$\"3AG/a2&F-7$Fh^l$\"3#og0R>?C0&F-7$Fb_l$\"35h?sc)H'Q ]F-7$Fg_l$\"3Cc_2)zu#G]F-7$F\\`l$\"3bg#3$R4v?]F-7$Fa`l$\"3X:*R!\\qh:]F -7$Ff`l$\"3n]URxdF6]F-7$F[al$\"3^o:5]kU3]F-7$F`al$\"3L$[1e,xh+&F-7$Fea l$\"3N@4eS))e/]F-7$Fjal$\"3WO`7kUL.]F--F_bl6&FablFbblFbblFcbl-F$6$7co7 $F($\"3Zzz\\dVmQ]F-7$F/$\"30iz%3R!H`]F-7$F5$\"3f6TtM7Yq]F-7$F:$\"3cZp \"Qm&['4&F-7$F?$\"3!ob$[\"GSC8&F-7$FD$\"3BC\"[<]*f\"=&F-7$FI$\"3oe!oy# GZV_F-7$FN$\"3-H'Q3))y+L&F-7$FS$\"3q1*[!H^l_aF-7$FX$\"3]LK#4^N5i&F-7$F gn$\"3`VL4AchheF-7$F\\o$\"3*fc0w:AC:'F-7$Fao$\"3*>%o'e'fv/mF-7$Ffo$\"3 #[%H'*fQ$*\\sF-7$F[p$\"3eV9)>jR'Q\")F-7$F`p$\"3)*H?oYry#G*F-7$Fep$\"3* e3O&3b1H6F17$Fjp$\"3#*[2,)))\\pQ\"F17$F_q$\"3E@(G'*=#Qj=F17$Fdq$\"3Kf% yM0X$QJ)))eF[en7$F\\im$\"3dG qU21>J5Fjen7$Faim$\"3UewO53#>D#Fjen7$Ffim$\"3JnD&onyt)QFjen7$F[jm$\"3i ;fRn&*Rd#)Fjen7$$!3*3]i:NLSo#Fbs$\"3VF`=q56)Q\"Fcgn7$F`jm$\"3UU5(=h*=0 GFcgn7$$!3A`7y]N4!\\\"Fbs$\"31y6h#3OP]%Fcgn7$$!3*R](=L?&zfF[en7$Fis$\"3IKg$GQ`K'GF[en7$F^t$\"3G2l3+** )Rn\"F[en7$Fht$\"3u5%ozj:,P(F17$Fbu$\"36(zFz\"\\/ySF17$F\\v$\"3w()=7dp XHEF17$Ffv$\"3&[:4v]en(=F17$F`w$\"3t&fqeCW**Q\"F17$F^y$\"3e'ek'QDq=6F1 7$F\\[l$\"3yAh6pbQ#G*F-7$Ff[l$\"3'3*)=#[(=u5)F-7$F`\\l$\"3p7:#\\?P*4sF -7$Fj\\l$\"31mAX6M]9mF-7$Fd]l$\"3M6nGic: w&fI3]F-7$F?$\"3\\4%G#)p)Q6]F-7$FD$\"3LQdpsFf:]F-7$FI$\"3\"osM)yk'3-&F -7$FN$\"39A,lDa@G]F-7$FS$\"38$>u%[=bQ]F-7$FX$\"3ar1&=!)GE0&F-7$Fgn$\"3 A#QF*p<]s]F-7$F\\o$\"3a:Q_kk:'4&F-7$Fao$\"3adb.> =?=&F-7$F[p$\"3N-s4Tu*yC&F-7$F`p$\"3SA2uY7RG`F-7$Fep$\"3u2]G]'*HfaF-7$ Fjp$\"3V7mVXA26cF-7$F_q$\"3K?Rxwla`eF-7$Fdq$\"3yk;ODGC]hF-7$Fiq$\"3he; S.Gl,mF-7$F^r$\"3g(>k[8Xi?(F-7$Fcr$\"3fqa4t&**R5)F-7$Fhr$\"3N^uzJ)RbG* F-7$F]s$\"30!H>aOG_6\"F17$Fds$\"3zT35Bo::9F17$F^t$\"3a1:aD>7F=F17$Fht$ \"3c!3Nf&H&[c#F17$Fbu$\"31HkyX()R$*RF17$F\\v$\"3rp>sc/kdqF17$Ffv$\"3Cp FWE1jM:F[en7$F[w$\"3cb$\\;]y\\$GF[en7$F`w$\"3eW0u9Xy3pF[en7$Few$\"3(o: s`U<_B\"Fjen7$Fjw$\"3*>^M!y]>6GFjen7$F_x$\"3pj[_7*4b0&Fjen7$Fdx$\"3WzD joC?k6Fcgn7$$\"3C(\\PfGcFy*F-$\"3v$)e**oh()=@Fcgn7$Fix$\"3Wiid[^'4+&Fc gn7$$\"3^Y(oHH5l*)*F-$\"3hGqNIS'pL*Fcgn7$$\"3F'\\7`HGW$**F-$\"3tnjHQ:x DBFahn7$$\"3.Yil(HYB(**F-$\"3')y?6jdl28Ffio7$F^y$\"3Jvd$H\\T;\\*Ffio7$ $\"3d\\P4')\\205F1$\"3:Ah6t#oE)QFahn7$$\"3c*\\(=UN745F1$\"3f*4-Dif8?\" Fahn7$$\"3a\\7G)4sJ,\"F1$\"3&f>1$e,bjdFcgn7$Fcy$\"3qQ+9IV0@Gh&*)Fjen7$F]z$\"38?. &>N)GjSFjen7$Fbz$\"3Tk)oovL%4BFjen7$Fgz$\"33WB$=+%=P5Fjen7$F\\[l$\"3g. 5!GFY\\'eF[en7$Fa[l$\"3x+)p!*Q(z\"z#F[en7$Ff[l$\"3=xtSQX9F;F[en7$F`\\l $\"3%>,B$o!)*=3(F17$Fj\\l$\"3^YkvLm6#3%F17$Fd]l$\"3=%Qfwd)=FEF17$F^^l$ \"3$fy/:,9>-(*)>6F17$Fg_l$ \"3xHFCWYw<$*F-7$F\\`l$\"3%3qV9$3&*e!)F-7$Fa`l$\"3\"QnBV9j\"[sF-7$Ff`l $\"3aO)olx08f'F-7$F[al$\"3u!pdV=?R<'F-7$F`al$\"39sw6Si&=&eF-7$Feal$\"3 wQ'RW>P$GcF-7$FjalF_bnF`bo-F$6%7S7$$!#5F*$!\"\"F*7$$!3\"pmmm\"p0k&*F1F a_q7$$!3vKL$3s%HaF1Fa_q7$$!3]******\\$*4)*\\F1Fa_q7$$!3p******\\_&\\c%F1Fa_q7$$!3$ )******\\1aZTF1Fa_q7$$!3Imm;/#)[oPF1Fa_q7$$!3%HLLL=exJ$F1Fa_q7$$!3lKLL L2$f$HF1Fa_q7$$!3$)****\\PYx\"\\#F1Fa_q7$$!3hLLLL7i)4#F1Fa_q7$$!3o)*** \\P'psm\"F1Fa_q7$$!3?****\\74_c7F1Fa_q7$$!3L:LL$3x%z#)F-Fa_q7$$!3()HLL 3s$QM%F-Fa_q7$$!3[^omm;zr)*F_sFa_q7$$\"3fVLLezw5VF-Fa_q7$$\"3..++v$Q# \\\")F-Fa_q7$$\"3%\\LL$e\"*[H7F1Fa_q7$$\"3>++++dxd;F1Fa_q7$$\"3f+++D0x w?F1Fa_q7$$\"34,+]i&p@[#F1Fa_q7$$\"3++++vgHKHF1Fa_q7$$\"3ElmmmZvOLF1Fa _q7$$\"3%4+++v+'oPF1Fa_q7$$\"3UKL$eR<*fTF1Fa_q7$$\"3L-++])Hxe%F1Fa_q7$ $\"3!fmm\"H!o-*\\F1Fa_q7$$\"3X,+]7k.6aF1Fa_q7$$\"3#emmmT9C#eF1Fa_q7$$ \"32****\\i!*3`iF1Fa_q7$$\"3 " 0 "" {MPLTEXT 1 0 0 " " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {XPPEDIT 18 0 "g2=0,g3=0" "6$/%#g2G\"\"!/%#g3GF%" }{TEXT -1 1 " " }} {PARA 14 "" 0 "" {TEXT -1 20 "The equation is now:" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 24 "Wp0:=subs(g2=0,g3=0,Wp);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$Wp0G,&*$)-%%diffG6$-%\"wG6#%\"zGF.\"\"#\"\"\"F0*& \"\"%F0)F+\"\"$F0!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 " R0 := differential_ring(ranking=[w],derivations=[z],notation=diff):\ne quations(Rosenfeld_Groebner([Wp0],R0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7#,&*$)-%%diffG6$-%\"wG6#%\"zGF.\"\"#\"\"\"F0*&\"\"%F0)F+\"\"$ F0!\"\"7#F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "equations(es sential_components(Wp0,R0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7#,& *$)-%%diffG6$-%\"wG6#%\"zGF.\"\"#\"\"\"F0*&\"\"%F0)F+\"\"$F0!\"\"" }}} {PARA 14 "" 0 "" {TEXT -1 33 "There is only one singular zero, " } {TEXT 25 6 "w(z)=0" }{TEXT -1 99 ", and it is not essential. We can se e its limiting property with respect to the non singular zeros." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "singsol:=plot(0,s=-4..4, co lor=red, thickness=2):\ngalsol:= plot(\{seq(1/((z-C)^2),C=-2..1)\},z=- 4..4):\nplots[display](\{singsol,galsol\},view=[-4..4,-1/2..2]);" }} {PARA 13 "" 1 "" {GLPLOT2D 399 227 227 {PLOTDATA 2 "6)-%'CURVESG6%7S7$ $!\"%\"\"!$F*F*7$$!3nmmmmFiDQ!#+++:v2*\\#F/F+7$$!3BLLL8>1DBF/F+7$$! 3lmmmw))yr@F/F+7$$!3:+++S(R#**>F/F+7$$!30++++@)f#=F/F+7$$!3-+++gi,f;F/ F+7$$!3qmmm\"G&R2:F/F+7$$!3WLLLtK5F8F/F+7$$!3fLLL$HsV<\"F/F+7$$!3+-++] &)4n**!#=F+7$$!37PLLL\\[%R)F]oF+7$$!3F)*****\\&y!pmF]oF+7$$!3Y******\\ O3E]F]oF+7$$!3NKLLL3z6LF]oF+7$$!3sLLL$)[`P(******z-6j'F]oF+7$$\"3q\"******4#32$)F]oF+7$$\"3r$*****\\#y'G**F] oF+7$$\"3H******H%=H<\"F/F+7$$\"35mmm1>qM8F/F+7$$\"3%)*******HSu]\"F/F +7$$\"3(HLL$ep'Rm\"F/F+7$$\"3()******R>4N=F/F+7$$\"3$emm;@2h*>F/F+7$$ \"3^*****\\c9W;#F/F+7$$\"3Lmmmmd'*GBF/F+7$$\"3j*****\\iN7]#F/F+7$$\"3a LLLt>:nEF/F+7$$\"36LLL.a#o$GF/F+7$$\"3ammm^Q40IF/F+7$$\"3z******z]rfJF /F+7$$\"3gmmmc%GpL$F/F+7$$\"3/LLL8-V&\\$F/F+7$$\"3>+++XhUkOF/F+7$$\"3= +++:o]Xy !\\i*oNF]o7$F4$\"3drT'y$H9DWF]o7$F7$\"3ZibEzhWScF]o7$F:$\"3XAj\"*3.WCu F]o7$F=$\"3f\\Sq+ule**F]o7$F@$\"35vr88M6C9F/7$FC$\"32U5(\\aW$RAF/7$FF$ \"3eqdo22![,%F/7$FI$\"3;)*Rl-&\\QY*F/7$$!3%******\\RD%[AF/$\"3eRiBspM? ;!#;7$FL$\"3k8G_i*=&)Q$F]x7$$!3!)****\\#f^'G@F/$\"3h5-i=i%=/'F]x7$$!3S LLL3V^&3#F/$\"3()H`RSX[n8!#:7$$!3U++Dmc%R1#F/$\"30b)**4(obXCF[y7$$!3+n m;CqPU?F/$\"3%G3-k26&obF[y7$$!3H+]7.FfJ?F/$\"3[1hNIZ!>+\"!#97$$!3eLL3# Q33-#F/$\"3[/gjZ(G&4BF[z7$$!3++Dc@iT:?F/$\"3yg!3jl$p2UF[z7$$!3(omT51C+ ,#F/$\"3ra^ET50_**F[z7$$!3tL3_+>j/?F/$\"3*yGB_HE5m%!#87$FO$\"3B*zW;(z6 IF/$\"3$enrQgOHi#F`[l7$$!3?++]P6T))>F/$\"3KuRgT-#fW (F[z7$$!3M++DOo*H)>F/$\"3`HeEi+#*eMF[z7$$!3D+++NDex>F/$\"367sF90))*)>F [z7$$!33++]KRvm>F/$\"3W'R%[]#4t/*F[y7$$!38+++I`#f&>F/$\"3V=]l@B!y9&F[y 7$$!3++++D\"oU$>F/$\"3#o4v$H3X9BF[y7$$!35+++?4h7>F/$\"3\\q9=(=S%48F[y7 $$!33+++5lHp=F/$\"3NOll0\\j`eF]x7$FR$\"3Qw2>hnE-LF]x7$$!3/+++!=*\\UGV?X$**F]o7$F_o$\"3C!)f' )4vcCuF]o7$Fbo$\"3'\\Xt)yb.FcF]o7$Feo$\"3LSNML=%*fWF]o7$Fho$\"3#))edBl 62f$F]o7$F[p$\"32b$RtAU$)*HF]o7$F^p$\"3$4b+X5,*4DF]o7$Fbp$\"3Ao'phi))) =@F]o7$Fep$\"35%)e4eHQ[=F]o7$Fhp$\"3os$>IZ`0h\"F]o7$F[q$\"3ECTV$R3+T\" F]o7$F^q$\"36me)GA')zC\"F]o7$Faq$\"3.D@d.JT;6F]o7$Fdq$\"3y4:!H-SI$**!# >7$Fgq$\"3**RE)GD9E**)F`bl7$Fjq$\"3eM4^C(o'G\")F`bl7$F]r$\"3wqH0C;)*[u F`bl7$F`r$\"3t+Yz7K0*z'F`bl7$Fcr$\"3))pM(\\u#=iiF`bl7$Ffr$\"3uO`F`bl7$F\\s$\"3sw4N(ygb$\\F`bl7$F_s$\"3-'3!4M=) 3f%F`bl7$Fbs$\"3JUkrQ&RWF%F`bl7$Fes$\"3EjF:vA'=*RF`bl7$Fhs$\"3(pa?#3() >cPF`bl7$F[t$\"3bI^bB,*3^$F`bl7$F^t$\"32ChLN`G6LF`bl7$Fat$\"3-s)p`#4l; JF`bl7$Fdt$\"3=vA`n&**f%HF`bl7$Fgt$\"3ixxxxxxxFF`bl-Fjt6&F\\u$\"#5!\" \"F+F+-F$6$7ao7$F($\"3066666666F]o7$F-$\"3\\zp4\"yzU Sl)R\"F]o7$F4$\"3H25eKa#ef\"F]o7$F7$\"3&G2>\"RUhR=F]o7$F:$\"3o7$obCLA9 #F]o7$F=$\"3gt>cYT#[\\#F]o7$F@$\"3UdDl$4>-'HF]o7$FC$\"3kUfqOf;$f$F]o7$ FF$\"3xhk>pg\"*\\WF]o7$FI$\"38,E/T\\W&p&F]o7$FL$\"3\"*pwI*>[GG(F]o7$FO $\"31\"=XdDA:+\"F/7$FR$\"3s(y?l.YdY\"F/7$FU$\"3#zCf([Ha-BF/7$FX$\"3%o. 0Mz\\U)QF/7$Fen$\"3![b%49c4Y$*F/7$$!3_LLL$yP2D\"F/$\"3U:va8!)f!f\"F]x7 $Fhn$\"3P?vP#Qd))G$F]x7$$!38++v$oc*H6F/$\"3zje/`\\5@fF]x7$$!3!pmmT2Tb3 \"F/$\"3)\\Od9!*GmO\"F[y7$$!3<+]PpKLj5F/$\"3$**Q1w#)zI\\#F[y7$$!3mLLek a7T5F/$\"3a=N8'=)f7fF[y7$$!3=+v=i:-I5F/$\"3(phVBj:&46F[z7$$!3$pm\"zfw \"*=5F/$\"3w12([Z\\Uz#F[z7$$!3>]Pf3dO85F/$\"3<]1'fa#y(f&F[z7$$!3oLeRdP \"y+\"F/$\"3Z3xDPF(yj\"F`[l7$$!3%p\"z>1=E-5F/$\"3#)ejl\"*4ua>!#77$F[o$ \"3%GRJH5HyB*F`[l7$$!3so\"zWPaz\"**F]o$\"36x!y3Qcb[\"F`[l7$$!3WN$e*)>5 )o)*F]o$\"3F^(y`M<.\"eF[z7$$!3<-vVBgm>)*F]o$\"39%o#[2$)*\\2$F[z7$$!3*) om\"z%=_q(*F]o$\"3xQp*z`m*)*=F[z7$$!3L-](o\\LAn*F]o$\"3s[&)=f0J3$*F[y7 $$!3yNL$e9XRd*F]o$\"3_^a[)p^*3bF[y7$$!3n-+vV%otP*F]o$\"3Wv_')QJ^zDF[y7 $$!3cpmmTwg3!\\\"F[y7$$!3M.+]P$Qwy)F]o$\"3a>pRq/b.oF]x 7$F_o$\"3W9Rgf&f%zQF]x7$$!3qnmmT+ %>$QI,*F/7$Feo$\"3Vx/o(pi?/%F/7$Fho$\"3a0Y4#**GbB#F/7$F[p$\"3IZ4\"Q13[ Y\"F/7$F^p$\"3epR#zXWz+\"F/7$Fbp$\"3$y@g,Yw[F(F]o7$Fep$\"3diM#Re]wo&F] o7$Fhp$\"3'*4#)eXXY$\\%F]o7$F[q$\"3lyNm0HT:OF]o7$F^q$\"3j*>woDXP)HF]o7 $Faq$\"33x#z.FEz^#F]o7$Fdq$\"3V+lcjv$z6#F]o7$Fgq$\"3#[\"Rw#)>eM=F]o7$F jq$\"3$\\I7Hw=0f\"F]o7$F]r$\"3L%\\?%y;549F]o7$F`r$\"3V!ygk$)HTC\"F]o7$ Fcr$\"3ZN(p-G+S6\"F]o7$Ffr$\"3_mC$Rr)\\')**F`bl7$Fir$\"3oTf\"\\HJO-*F` bl7$F\\s$\"3CSjb3X]d\")F`bl7$F_s$\"3=8z6#)z/OuF`bl7$Fbs$\"3Kj76F5\"Hz' F`bl7$Fes$\"3Zh`,p?6MiF`bl7$Fhs$\"3hd0#G&zEzdF`bl7$F[t$\"39y)\\\"G$=mJ &F`bl7$F^t$\"34ZYj8mJ[\\F`bl7$Fat$\"3RNE&>5\\if%F`bl7$Fdt$\"3kEO$GzALH %F`bl7$Fgt$\"33+++++++SF`bl-Fjt6&F\\uF+FfelF+-F$6$7co7$F($\"3+++++++]i F`bl7$F-$\"3v_.+sGvKoF`bl7$F1$\"3A;4EuXw3uF`bl7$F4$\"3$QIJVNV![\")F`bl 7$F7$\"3(pn!=eL()4!*F`bl7$F:$\"3NpR#>J'3,5F]o7$F=$\"3xqd]Ind46F]o7$F@$ \"3W6')H))*3;C\"F]o7$FC$\"3)Ql%*fj!e/9F]o7$FF$\"3:OKnM9=,;F]o7$FI$\"3! y[&*fKE)\\=F]o7$FL$\"3O.%>nAT,7#F]o7$FO$\"3vq1HM]#F]o7$FR$\"39v?=T1 @**HF]o7$FU$\"3=ClC<(yKj$F]o7$FX$\"3fEfsDD%4S%F]o7$Fen$\"3$Q7rk'o$zn&F ]o7$Fhn$\"3;p\"*y?9%3D(F]o7$F[o$\"3r%e9#3Hh15F/7$F_o$\"3a>9fzj4>9F/7$F bo$\"3A=%3='GP[AF/7$Feo$\"3_W2Zu/feRF/7$Fho$\"3=..L4qY<\"*F/7$$!3/LLLe GmCDF]o$\"3X62s7E*)o:F]x7$F[p$\"3v%*=hW\\K7LF]x7$$!39nm;aH-88F]o$\"3T] iFJbO+eF]x7$$!3l0++]-6&)))F`bl$\"3=sr(Hr,nE\"F[y7$$!3vsm;/1binF`bl$\"3 FM.dO$[m=#F[y7$$!3&)RLLe4**RYF`bl$\"3dEP9NiyWYF[y7$$!3TtmTN6ryNF`bl$\" 3SEi0iT73yF[y7$$!3'p++DJJu^#F`bl$\"3MG=gE\">zd\"F[z7$$!3tt;/,9z')>F`bl $\"3\"\\KNx_^L`#F[z7$$!3^SLe*[^hX\"F`bl$\"3Cf/L-@9;ZF[z7$$!3%G2]7y:^D* F`p$\"3By_$*=YWn6F`[l7$F^p$\"3]7TanfR8kF`[l7$$\"3BTK3x19j:F`p$\"3r-Ovr Ik#4%Fd\\m7$$\"3/BK$3-)*\\2(F`p$\"3CHBG!fyx*>F`[l7$$\"3\\?$ek`&oe7F`bl $\"3AS5J\"ey>J'F[z7$$\"3n=L$3Fr)4=F`bl$\"3@1iL0T%G0$F[z7$$\"3.:LeRFC7H F`bl$\"3K&3fEI%3z6F[z7$$\"3S6LL3Uh9SF`bl$\"3W\\Wdo'zX?'F[y7$$\"3w2L3xc )p6&F`bl$\"3?L:`EH>>QF[y7$$\"37/L$e9d$>iF`bl$\"3v_TF_%)G&e#F[y7$$\"3\\ +Le9'GLSl=F[y7$$\"3&oHLL3+TU)F`bl$\"3y$ei_uAF/7$F^q$\"3^*)*QiZ:\"\\9F/7$Faq$\"3U8d+=%=W,\"F/7$ Fdq$\"37.ndzx#)osF]o7$Fgq$\"38P++%)f\"F]o7$F_s$\"3`@w \"GHQdS\"F]o7$Fbs$\"3O[aOJ)4EC\"F]o7$Fes$\"3cAS1wuM26F]o7$Fhs$\"3Cd'z, pA;+\"F]o7$F[t$\"3*y&3V'p<1)*)F`bl7$F^t$\"3G83a;Ri%=)F`bl7$Fat$\"3Pv$3 x)[6ZuF`bl7$Fdt$\"31j,+'=u2$oF`bl7$FgtFjem-Fjt6&F\\uFfelFfelF+-F$6$7go 7$F(Fbem7$F-$\"3l\\gF:(3VH%F`bl7$F1$\"3$Qcz>&)Qwd%F`bl7$F4$\"3)[%*QYL2 6$\\F`bl7$F7$\"3:\\:G_p$*H`F`bl7$F:$\"3Lo%4)zo\"px&F`bl7$F=$\"3(o'H(\\ :DNC'F`bl7$F@$\"3kM)R$=^'))y'F`bl7$FC$\"3%3aPMP$fJuF`bl7$FF$\"3eQ]*\\9 qv;)F`bl7$FI$\"3IR:frC$[/*F`bl7$FL$\"3Qo(Q04;,%**F`bl7$FO$\"3NW8\"3[u; 6\"F]o7$FR$\"3r`?YAR;_7F]o7$FU$\"3w#[JWk`VT\"F]o7$FX$\"30L7[wed!f\"F]o 7$Fen$\"3UB\"pVP#eY=F]o7$Fhn$\"3Q0]M_i5:@F]o7$F[o$\"3!G?)3/dC3DF]o7$F_ o$\"3P!GEYida&HF]o7$Fbo$\"3=u$GDHe*)f$F]o7$Feo$\"3z09$3pF!HWF]o7$Fho$ \"3b\"RU2j?Kk&F]o7$F[p$\"3C+N**\\()[esF]o7$F^p$\"3SI=(o)4\\@**F]o7$Fbp $\"3M\"ypp;H,Y\"F/7$Fep$\"3*Qj$yZf5,AF/7$Fhp$\"3u;Vw#G#*=(QF/7$F[q$\"3 5h&H2T&)4\"))F/7$$\"3W%*****\\C4puF]o$\"3+Pe2z+;h:F]x7$F^q$\"3JOb_<)=# *[$F]x7$$\"3&>**\\(=;y4&)F]o$\"32I$\\fb\")H]%F]x7$$\"3?#***\\P6[7()F]o $\"3__&)*31PC.'F]x7$$\"3X#**\\il!=:*)F]o$\"3i)*pW>BQ(\\)F]x7$$\"3q#*** *\\+WF[y7$$\"3L$*\\(=(*GYi*F]o$\"3?8]A#Qgq 4(F[y7$$\"3Y$**\\7tyfs*F]o$\"36gbWGoxJ8F[z7$$\"3_$\\P4h`mx*F]o$\"3^\"G $R#[mY+#F[z7$$\"3e$*\\i!\\Gt#)*F]o$\"3-20<'zlRN$F[z7$$\"3l$\\7.P.!y)*F ]o$\"3qzH/__**=nF[z7$Faq$\"3)[S5q`ve'>F`[l7$$\"38$\\iS1W\\)**F]o$\"3U? pq;<[6WFd\\m7$$\"3D*\\7y)47/5F/$\"3;Ns2%[+%))eF`[l7$$\"3U\\(=#pvu45F/$ \"3qZWmHUY_5F`[l7$$\"3O**\\i]TP:5F/$\"3r+A2ROvIUF[z7$$\"3I\\7.K2+@5F/$ \"3RzQ)ohIUFF[y7$$\"3L****\\FJ*G3\"F/$\"3(Hd'o:\\Lb9F[y7$ $\"3K**\\7`%*R06F/$\"3pJ&4*)3u;+*F]x7$$\"3J***\\(yd!z7\"F/$\"3Bwa!\\>6 D6'F]x7$$\"3I**\\P/@T]6F/$\"3@Xj0?/WTRV\"F/7$Fcr$\"3PGa=v7$y+\" F/7$Ffr$\"3(pVm8y&QvtF]o7$Fir$\"3#yNst?L?m&F]o7$F\\s$\"3p'3;)f78PWF]o7 $F_s$\"3]Bjb!R/zf$F]o7$Fbs$\"3!oJ'HrZ!R'HF]o7$Fes$\"3QUpPdQJ([#F]o7$Fh s$\"3\"3Lf$4E\"R9#F]o7$F[t$\"3BSf9(y(3J=F]o7$F^t$\"36lp@/a'eg\"F]o7$Fa t$\"3IjD\"F]o7$FgtF]fl-Fjt6&F\\uF+F+Ffel- %+AXESLABELSG6$Q\"s6\"Q!6\"-%%VIEWG6$;F(Fgt;$!0+++++++&F[y$FcuF*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " "Curve 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}} {MARK "33 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }