A technique for solving ordinary differential equations using Riemann's P-functions

Abstract

This paper presents an algorithmic approach to symbolic solution of 2nd order linear ODEs. The algorithm consists of two parts. The first part involves complete algorithms for hypergeometric equations and hypergeometric equations of confluent type. These algorithms are based on Riemann's P-functions and Hukuhara's P-functions respectively. Another part involves an algorithm for transforming a given equation to a hypergeometric equation or a hypergeometric equation of confluent type. The transformation is possible if a given equation satisfies certain conditions, otherwise it works only as one of heuristic methods. However our method can solve many equations which seem to be very difficult to solve by conventional methods.