A geometric approach to integrability of Abel differential equations

TL;DR

This paper introduces a geometric method based on quasi-Lie systems to analyze the integrability of Abel differential equations, including second order cases and their Lagrangian formulations, using Darboux polynomials and Jacobi multipliers.

Contribution

It presents a novel geometric framework for understanding Abel equations' integrability and explores non-natural Lagrangian formulations for second order cases.

Findings

Characterization of integrable Abel equations using quasi-Lie systems

Existence of two non-natural Lagrangian formulations for second order Abel equations

Application of Darboux polynomials and Jacobi multipliers in the analysis

Abstract

A geometric approach is used to study the Abel first order differential equation of the first kind. The approach is based on the recently developed theory of quasi-Lie systems which allows us to characterise some particular examples of integrable Abel equations. Second order Abel equations will be discussed and the inverse problem of the Lagrangian dynamics is analysed: the existence of two alternative Lagrangian formulations is proved, both Lagrangians being of a non-natural class. The study is carried out by means of the Darboux polynomials and Jacobi multipliers.