|L||A second order differential operator|
Darboux(L) returns Darboux polynomials for of lowest possible degree. Note that is a solution of if and only if for some Darboux polynomial .
If Darboux(L) returns , then this proves that is irreducible and has no nontrivial Darboux curves, and hence that has no Liouvillian solution.
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
> L := D^2 + 3/16/x^2 + 2/9/(x-1)^2 - 3/(16*x*(x-1)); > Darboux(L, D, x, u);