Abel ODEs: Equivalence and Integrable Classes

TL;DR

This paper classifies non-constant invariant Abel ODEs using invariant theory, introduces new integrable classes, and develops computer algebra routines for systematic solving within Maple.

Contribution

It provides a new classification of Abel ODEs, identifies novel integrable classes, and creates computational tools for solving these equations systematically.

Findings

Classification of Abel ODEs based on invariants

Introduction of new integrable classes depending on parameters

Development of Maple routines for solving Abel ODEs

Abstract

A classification, according to invariant theory, of non-constant invariant Abel ODEs known as solvable and found in the literature is presented. A set of new integrable classes depending on one or no parameters, derived from the analysis of the works by Abel, Liouville and Appell, is also shown. Computer algebra routines were developed to solve ODEs members of these classes by solving their related equivalence problem. The resulting library permits a systematic solving of Abel type ODEs in the Maple symbolic computing environment.