Dealing with Rational Second Order Ordinary Differential Equations where both Darboux and Lie Find It Difficult: The $S$-function Method

TL;DR

This paper introduces a new method called the S-function approach for finding first integrals of rational second order ODEs, which often succeeds where Darbouxian and symmetry methods fail, supported by a Maple implementation.

Contribution

The paper presents a novel S-function method for solving rational second order ODEs, providing an alternative to traditional Darbouxian and symmetry techniques.

Findings

The S-function method successfully finds first integrals in cases where Darbouxian and symmetry methods do not.

Implementation of the method in the Maple package { extit{InSyDE}} facilitates analysis of the solution process.

The approach broadens the toolkit for solving complex rational second order ODEs.

Abstract

Here we present a new approach to search for first order invariants (first integrals) of rational second order ordinary differential equations. This method is an alternative to the Darbouxian and symmetry approaches. Our procedure can succeed in many cases where these two approaches fail. We also present here a Maple implementation of the theoretical results and methods, hereby introduced, in a computational package -- {\it InSyDE}. The package is designed, apart from materializing the algorithms presented, to provide a set of tools to allow the user to analyse the intermediary steps of the process.