A class of exact solutions of the Liénard type ordinary non-linear differential equation

Abstract

A class of exact solutions is obtained for the Liénard type ordinary non-linear differential equation. As a first step in our study the second order Liénard type equation is transformed into a second kind Abel type first order differential equation. With the use of an exact integrability condition for the Abel equation (the Chiellini lemma), the exact general solution of the Abel equation can be obtained, thus leading to a class of exact solutions of the Liénard equation, expressed in a parametric form. We also extend the Chiellini integrability condition to the case of the general Abel equation. As an application of the integrability condition the exact solutions of some particular Liénard type equations, including a generalized van der Pol type equation, are explicitly obtained.

18 pages, no figures; minor revisions, accepted for publication in Journal of Engineering Mathematics

References in corpus (3)

• Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations
• A nonlocal connection between certain linear and nonlinear ordinary differential equations/oscillators
• A Chiellini type integrability condition for the generalized first kind Abel differential equation

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• Quasi-Lie schemes for PDEs
• On the integrability of the Abel and of the extended Liénard equations
• Alternative Lagrangians obtained by scalar deformations
• Isochronous waveforms of Liénard equations via commutative factorization