Linearizable ordinary differential equations

TL;DR

This paper investigates conditions under which planar polynomial differential systems can be linearized using inverse integrating factors derived from solutions of linear differential equations, including families related to orthogonal polynomials.

Contribution

It introduces new families of differential systems where integrability is achieved via solutions of linear equations, expanding understanding of linearization methods.

Findings

Identification of systems linearizable through linear differential equations

Construction of inverse integrating factors using solutions of second-order linear equations

Connection between Darboux integrability and orthogonal polynomial solutions

Abstract

Our purpose in this paper is to study when a planar differential system polynomial in one variable linearizes in the sense that it has an inverse integrating factor which can be constructed by means of the solutions of linear differential equations. We give several families of differential systems which illustrate how the integrability of the system passes through the solutions of a linear differential equation. At the end of the work, we describe some families of differential systems which are Darboux integrable and whose inverse integrating factor is constructed using the solutions of a second--order linear differential equation defining a family of orthogonal polynomials.