Polynomial solutions of certain differential equations arising in physics

TL;DR

This paper develops criteria and an algorithm to find polynomial solutions of linear differential equations with polynomial coefficients, which are common in physics, and demonstrates their effectiveness through applications in mathematical physics.

Contribution

It provides necessary and sufficient conditions and an algorithm for constructing polynomial solutions of such differential equations, advancing methods in mathematical physics.

Findings

Derived conditions for polynomial solutions of differential equations

Presented an algorithm for solution construction

Applied the method to physics-related equations

Abstract

Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An algorithm to determine these conditions and to construct the polynomial solutions is given. The effectiveness of this algorithmic approach is illustrated by applying it to several differential equations that arise in mathematical physics.