Integrable Abel equations and Vein's Abel equation

TL;DR

This paper advances the understanding of Abel equations by reformulating key integrability results, applying them to Vein's Abel equation, and analyzing its solutions and phase space behavior.

Contribution

It introduces new findings in Abel equation integrability and applies these to Vein's Abel equation, expressing solutions via third order hyperbolic functions.

Findings

New integrability results for Abel equations

Explicit solutions of Vein's Abel equation using hyperbolic functions

Phase space analysis of the nonlinear oscillator

Abstract

We first reformulate and expand with several novel findings some of the basic results in the integrability of Abel equations. Next, these results are applied to Vein's Abel equation whose solutions are expressed in terms of the third order hyperbolic functions and a phase space analysis of the corresponding nonlinear oscillator is also provided