General Solutions of the Abel Differential Equations

TL;DR

This paper introduces a new derivative condition for solving first-kind Abel differential equations, enabling the derivation of general solutions for both first and second kinds, with potential extensions to Lienard equations.

Contribution

It presents a novel derivative condition for first-kind Abel equations and derives their general solutions, also linking to second-kind solutions and Lienard equations.

Findings

Derived general solutions for first-kind Abel equations with zero free term

Extended solutions to second-kind Abel equations

Discovered a pair of entangled functions

Abstract

The Abel differential equations play a significant role in various fields of mathematics and applied sciences and are classified into two types: the first kind and the second kind. A novel derivative condition for the general solution of first-kind Abel equation is introduced. Based on this condition, the general solutions to the first-kind Abel equation with a zero free term are obtained, which in turn enables the derivation of the general solutions to the second-kind Abel equation, and meanwhile, a pair of entangled functions is discovered. These results can be extended to the Lienard equation.