Symbolic analysis of second-order ordinary differential equations with polynomial coefficients

TL;DR

This paper presents a symbolic computational method using the theta-operator to analyze the singularity structure of second-order linear ODEs with polynomial coefficients, facilitating solutions in terms of special functions.

Contribution

It introduces the symODE2 package that automates the analysis of singularities and solutions for hypergeometric and Heun-type equations using SageMath.

Findings

The theta-operator method effectively derives indicial equations and recurrence relations.

The symODE2 package automates the symbolic analysis of second-order ODEs with polynomial coefficients.

The approach simplifies obtaining solutions in terms of special functions for relevant differential equations.

Abstract

The singularity structure of a second-order ordinary differential equation with polynomial coefficients often yields the type of solution. It is shown that the $\theta$-operator method can be used as a symbolic computational approach to obtain the indicial equation and the recurrence relation. Consequently, the singularity structure leads to the transformations that yield a solution in terms of a special function, if the equation is suitable. Hypergeometric and Heun-type equations are mostly employed in physical applications. Thus only these equations and their confluent types are considered with SageMath routines which are assembled in the open-source package symODE2.