A semi-algorithm to find elementary first order invariants of rational second order ordinary differential equations

TL;DR

This paper introduces a semi-algorithmic method for identifying elementary first integrals of rational second order ODEs, enabling determination and computation of solutions using Darboux-type procedures.

Contribution

It presents a novel semi-algorithm that detects and computes elementary first integrals of rational SOODEs up to a certain polynomial degree.

Findings

Method can determine if a rational SOODE has an elementary first integral.

The approach finds the integral via quadratures when it exists.

Practical computational considerations are addressed.

Abstract

Here we present a method to find elementary first integrals of rational second order ordinary differential equations (SOODEs) based on a Darboux type procedure \cite{ManMac,firsTHEOps1,secondTHEOps1}. Apart from practical computational considerations, the method will be capable of telling us (up to a certain polynomial degree) if the SOODE has an elementary first integral and, in positive case, finds it via quadratures.