Non-Liouvillian solutions for second order linear ODEs

TL;DR

This paper introduces an algorithm to compute special function solutions for second order linear ODEs, extending beyond Liouvillian solutions, and successfully solves 91% of a standard set of equations.

Contribution

It presents a new, implementable algorithm for finding hypergeometric function solutions to second order linear ODEs, covering cases not solvable by Liouvillian methods.

Findings

Successfully solves 91% of Kamke's second order ODE examples

Extends solution methods to include hypergeometric functions

Provides an easy-to-implement algorithm in computer algebra systems

Abstract

There exist sound literature and algorithms for computing Liouvillian solutions for the important problem of linear ODEs with rational coefficients. Taking as sample the 363 second order equations of that type found in Kamke's book, for instance, 51 % of them admit Liouvillian solutions and so are solvable using Kovacic's algorithm. On the other hand, special function solutions not admitting Liouvillian form appear frequently in mathematical physics, but there are not so general algorithms for computing them. In this paper we present an algorithm for computing special function solutions which can be expressed using the 2F1, 1F1 or 0F1 hypergeometric functions. The algorithm is easy to implement in the framework of a computer algebra system and systematically solves 91 % of the 363 Kamke's linear ODE examples mentioned.