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

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

• When using Darboux from inside MAPLE, you must give the symbol for the returned curves as an extra argument. So the above example in MAPLE would be:

  > L := D^2 + 3/16/x^2 + 2/9/(x-1)^2 - 3/(16*x*(x-1));  
  > Darboux(L, D, x, u);
  

  $$\begin{eqnarray*} \left[ \left ({x}^{12}\right.\right. &-&\left. 6\,{x}^{11}+15\... ...36864}}\,{x}^{2}-{\frac {41}{4096}}\,x+{\frac {3}{4096}} \right] \end{eqnarray*}$$