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, issue 1

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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https://doi.org/10.1155/2012/171205

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnljam:v:2012:y:2012:i:1:n:171205

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