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Darboux


Usage

Darboux L


Parameter Type Description
L $\mathbbm{Q}[x,\frac d{dx}]$ A second order differential operator


Returns

Darboux(L) returns Darboux polynomials for $L$ of lowest possible degree. Note that $y = e^{\int u dx}$ is a solution of $L y = 0$ if and only if $p(x,u) = 0$ for some Darboux polynomial $p$.


Remarks

If Darboux(L) returns $[]$, then this proves that $L$ is irreducible and has no nontrivial Darboux curves, and hence that $L y = 0$ has no Liouvillian solution.


Example

To look for closed-form solutions of the differential equation

\begin{displaymath}
\frac{d^2 y}{dx^2} +
{\left({\frac 3{16 x^2} + \frac 2{9 (x-1)^2} - \frac 3{16 x (x-1)}}\right)} y = 0
\end{displaymath} (1)

we look for a Darboux polynomial as follows:
1 --> L := D^2 + 3/16/x^2 + 2/9/(x-1)^2 - 3/(16*x*(x-1));  
2 --> v := Darboux(L);
3 --> tex(v);

\begin{eqnarray*}[ u^{6}&+&{{-4\,x+2} \over {x^{2}-x}}\,u^{5}+
{{{{20} \over {3}...
...2}-6\,x^{11}+15\,x^{10}-20\,x^{9}+15\,x^{8}-
6\,x^{7}+x^{6}}} ]
\end{eqnarray*}



This means that (1) has a solution $y(x)$ whose logarithmic derivative is a root of the above polynomial.


Usage within MAPLE


next up previous contents index
Next: decompose Up: Supported functions Previous: coefficient   Contents   Index
Manuel Bronstein 2002-09-04