On connections of the Li\'enard equation with some equations of Painlev\'e--Gambier type

Abstract

The Li\'enard equation is used in various applications. Therefore, constructing general analytical solutions of this equation is an important problem. Here we study connections between the Li\'enard equation and some equations from the Painlev\'e--Gambier classification. We show that with the help of such connections one can construct general analytical solutions of the Li\'enard equation's subfamilies. In particular, we find three new integrable families of the Li\'enard equation. We also propose and discuss an approach for finding one--parameter families of closed--form analytical solutions of the Li\'enard equation.