Improving a family of Darboux methods for rational second order ordinary differential equations

TL;DR

This paper introduces algorithms that significantly reduce the computational effort in finding Darboux polynomials, which are essential for solving second order ordinary differential equations using Darboux methods.

Contribution

The paper presents new algorithms based on the structure of second order ODEs and differential invariants to efficiently determine Darboux polynomials, improving existing methods.

Findings

Algorithms reduce computation time for Darboux polynomial determination.

Structural analysis of second order ODEs informs the new algorithms.

A theorem relating differential invariants to Darboux polynomials is established.

Abstract

We have been working in many aspects of the problem of analyzing, understanding and solving ordinary differential equations (first and second order). As we have extensively mentioned, while working in the Darboux type methods, the most costly step of our methods and algorithms of solution is the determination of Darboux polynomials for the associated differential operators. Here, we are going to present some algorithms to greatly reduce the time expenditure in determining these needed Darboux polynomials. Some of them are based on a detailed analysis of the general structure of second order differential equations regarding the associated differential invariants. In order to perform this analysis, we produce a theorem concerning the general form for the differential invariants in terms of the Darboux polynomials.