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

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);