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### Darboux

Usage

Darboux L

Parameter Type Description
L A second order differential operator

Returns

Darboux(L) returns Darboux polynomials for of lowest possible degree. Note that is a solution of if and only if for some Darboux polynomial .

Remarks

If Darboux(L) returns , then this proves that is irreducible and has no nontrivial Darboux curves, and hence that has no Liouvillian solution.

Example

To look for closed-form solutions of the differential equation

 (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);


This means that (1) has a solution 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);


Next: decompose Up: Supported functions Previous: coefficient   Contents   Index
Manuel Bronstein 2002-09-04