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A class of exact solutions of the Liťnard-type ordinary nonlinear differential equation

T. Harko, F. S. N. Lobo, M. K. Mak

Abstract
A class of exact solutions is obtained for the Liénard-type ordinary nonlinear differential equation. As a first step in our study, the second-order Liénard-type equation is transformed into a first Abel kind, first order differential equation. With the use of an exact integrability condition for the Abel equation (Chiellini lemma), the exact general solution of the Abel equation can be obtained, leading to a class of exact solutions of the Liénard equation, expressed in 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.

Keywords
Abel equation - Exact solutions - Integrability condition - Liťnard equation

Journal of Engineering Mathematics
Volume 89, Number 1, Page 193
2014 December

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Instituto de Astrof√≠sica e Ci√™ncias do Espa√ßo Universidade do Porto Faculdade de Ciências da Universidade de Lisboa
Fundação para a Ciência e a Tecnologia COMPETE 2020 PORTUGAL 2020 União Europeia