An Abel ordinary differential equation class generalizing known integrable classes

TL;DR

This paper introduces a multi-parameter class of Abel differential equations that unifies many known integrable classes, includes new subclasses, and systematically relates to Riccati equations, expanding the landscape of integrable ODEs.

Contribution

It presents a comprehensive multi-parameter class of Abel ODEs that unifies existing integrable classes and introduces new integrable subclasses with systematic mappings to Riccati equations.

Findings

Unifies all known integrable Abel classes

Includes new fully integrable subclasses

Systematically maps many members to Riccati equations

Abstract

We present a multi-parameter non-constant-invariant class of Abel ordinary differential equations with the following remarkable features. This one class is shown to unify, that is, contain as particular cases, all the integrable classes presented by Abel, Liouville and Appell, as well as all those shown in Kamke's book and various other references. In addition, the class being presented includes other new and fully integrable subclasses, as well as the most general parameterized class of which we know whose members can systematically be mapped into Riccati equations. Finally, many integrable members of this class can be systematically mapped into an integrable member of a different class. We thus find new integrable classes from previously known ones.