The rational solutions of a holonomic system form a finite-dimensional vector space. The only possibilities for the poles of a rational solution are the codimension one components of the singular locus. An algorithm to compute rational solutions is based on Gr\"obner deformations and works for ideals $I$ of PDE's - see the paper Polynomial and rational solutions of a holonomic system by Oaku, Takayama and Tsai (2000).
i1 : makeWA(QQ[x]) o1 = QQ[x, dx] o1 : PolynomialRing, 1 differential variables |
i2 : I = ideal((x+1)*dx+5) o2 = ideal(x*dx + dx + 5) o2 : Ideal of QQ[x, dx] |
i3 : RatSols I 1 o3 = {-------------------------------} 5 4 3 2 x + 5x + 10x + 10x + 5x + 1 o3 : List |
The most efficient method to find rational solutions of a system of differential equations is to find the singular locus, then try to find its irreducible factors. With these, call RatSols(I, ff, w), where w should be generic enough so that the PolySols routine will not complain of a non-generic weight vector.
The object RatSols is a method function.