On connections of the Liénard equation with some equations of Painlevé--Gambier type

Abstract

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

References in corpus (5)

• Analytical solutions for problems of bubble dynamics
• Analytical solutions of the Rayleigh equation for empty and gas--filled bubble
• On the criteria for integrability of the Liénard equation
• Linearization of Second-Order Ordinary Differential Equations by Generalized Sundman Transformations
• A note on existence and uniqueness of limit cycles for Liénard systems

Cited by in corpus (1)

• On linearizability via nonlocal transformations and first integrals for second-order ordinary differential equations