Solving higher order linear differential equations having elliptic function coefficients

Abstract

We consider the problem of finding closed form solutions of a linear homogeneous ordinary differential equation having coefficients which are elliptic functions. In particular, the input coefficients are assumed to be represented as elements of C(p,p'), where C is the complex number field, and p(x) and p' (x) are the Weierstrass p function and its first derivative, respectively. The specific closed form solutions y(x) which we seek are hyperexponential over C(p,p'), i.e., solutions y(x) such that y' (x)/y(x) is in C(p,p'). Such solutions correspond to first order right-hand factors of the associated linear differential operator, and are analogous to hyperexponential solutions over C(x), in the more well-known case where the coefficients of the ode are in C(x). A previous paper [4] gave an algorithm for equations of second order. The algorithm presented here works for equations of arbitrary order, and will find all such hyperexponential solutions that may exist. It relies on determining the structure of such first order factors to construct an ansatz of a solution, which can then be completely determined by solving a system of multivariate polynomial equations. The algorithm works well for solutions having few singularities and hidden poles, but can slow as the number of such points increases.