Lie and Riccati Linearization of a Class of Liénard Type Equations

A. G. Johnpillai, C. M. Khalique and F. M. Mahomed

Journal of Applied Mathematics, 2012, vol. 2012, 1-8

Abstract:

We construct a linearizing Riccati transformation by using an ansatz and a linearizing point transformation utilizing the Lie point symmetry generators for a three-parameter class of Liénard type nonlinear second-order ordinary differential equations. Since the class of equations also admits an eight-parameter Lie group of point transformations, we utilize the Lie-Tresse linearization theorem to obtain linearizing point transformations as well. The linearizing transformations are used to transform the underlying class of equations to linear third- and second-order ordinary differential equations, respectively. The general solution of this class of equations can then easily be obtained by integrating the linearized equations resulting from both of the linearization approaches. A comparison of the results deduced in this paper is made with the ones obtained by utilizing an approach of mapping the class of equations by a complex point transformation into the free particle equation. Moreover, we utilize the linearizing Riccati transformation to extend the underlying class of equations, and the Lie-Tresse linearization theorem is also used to verify the conditions of linearizability of this new class of equations.

Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnljam:171205

DOI: 10.1155/2012/171205

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