General Solutions of the Abel Differential Equations

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.