Finding Elementary First Integrals for Rational Second Order Ordinary Differential Equations

TL;DR

This paper introduces a new algorithm for finding elementary first integrals of rational second order ODEs, offering a more robust theoretical foundation and broader applicability than previous methods.

Contribution

The paper presents a theoretically grounded, more comprehensive algorithm for solving rational second order ODEs without restricted assumptions, enabling faster and easier integrability analysis.

Findings

Broader class of SOODEs covered

Faster computation of first integrals

Enhanced theoretical basis with new theorems

Abstract

Here we present an algorithm to find elementary first integrals of rational second order ordinary differential equations (SOODEs). In \cite{PS2}, we have presented the first algorithmic way to deal with SOODEs, introducing the basis for the present work. In \cite{royal}, the authors used these results and developed a method to deal with SOODEs and a classification of those. Our present algorithm is based on a much more solid theoretical basis (many theorems are presented) and covers a much broader family of SOODEs than before since we do not work with restricted ansatz. Furthermore, our present approach allows for an easy integrability analysis of SOODEs and much faster actual calculations.