scholarly journals Lie and Riccati Linearization of a Class of Liénard Type Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-8
Author(s):  
A. G. Johnpillai ◽  
C. M. Khalique ◽  
F. M. Mahomed
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Free Particle ◽  
Second Order ◽  
Point Transformation ◽  
Riccati Transformation ◽  
Linearization Theorem ◽  
Point Transformations ◽  
Symmetry Generators ◽  
Comparison Of The Results

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.

scholarly journals Some Remarks on the Solution of Linearisable Second-Order Ordinary Differential Equations via Point Transformations

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Winter Sinkala
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lie Group ◽  
Second Order ◽  
Point Transformation ◽  
Group Method ◽  
Lie Group Method ◽  
Generic Solution ◽  
Point Transformations

Transformations of differential equations to other equivalent equations play a central role in many routines for solving intricate equations. A class of differential equations that are particularly amenable to solution techniques based on such transformations is the class of linearisable second-order ordinary differential equations (ODEs). There are various characterisations of such ODEs. We exploit a particular characterisation and the expanded Lie group method to construct a generic solution for all linearisable second-order ODEs. The general solution of any given equation from this class is then easily obtainable from the generic solution through a point transformation constructed using only two suitably chosen symmetries of the equation. We illustrate the approach with three examples.

scholarly journals Linearization of Two Second-Order Ordinary Differential Equations via Fiber Preserving Point Transformations

2011 ◽  
Vol 2011 ◽  
pp. 1-21 ◽  
Author(s):  
Sakka Sookmee ◽  
Sergey V. Meleshko
Keyword(s):  
Linear System ◽  
General Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Necessary Conditions ◽  
Second Order ◽  
Point Transformations ◽  
Constant Coefficients ◽  
Linearizable System

The necessary form of a linearizable system of two second-order ordinary differential equations y1″=f1(x,y1,y2,y1′,y2′), y2″=f2(x,y1,y2,y1′,y2′) is obtained. Some other necessary conditions were also found. The main problem studied in the paper is to obtain criteria for a system to be equivalent to a linear system with constant coefficients under fiber preserving transformations. A linear system with constant coefficients is chosen because of its simplicity in finding the general solution. Examples demonstrating the procedure of using the linearization theorems are presented.

scholarly journals On the complete integrability and linearization of certain second-order nonlinear ordinary differential equations

2005 ◽  
Vol 461 (2060) ◽  
pp. 2451-2477 ◽  
Author(s):  
V.K Chandrasekar ◽  
M Senthilvelan ◽  
M Lakshmanan
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Free Particle ◽  
First Integrals ◽  
Second Order ◽  
Complete Integrability ◽  
Nonlinear Ordinary Differential Equations ◽  
Integrals Of Motion ◽  
Integrating Factors

A method for finding general solutions of second-order nonlinear ordinary differential equations by extending the Prelle–Singer (PS) method is briefly discussed. We explore integrating factors, integrals of motion and the general solution associated with several dynamical systems discussed in the current literature by employing our modifications and extensions of the PS method. We also introduce a novel way of deriving linearizing transformations from the first integrals to linearize the second-order nonlinear ordinary differential equations to free particle equations. We illustrate the theory with several potentially important examples and show that our procedure is widely applicable.

scholarly journals Linearization from Complex Lie Point Transformations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Sajid Ali ◽  
M. Safdar ◽  
Asghar Qadir
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lie Algebras ◽  
Complex Structure ◽  
Second Order ◽  
Geometrical Construction ◽  
Maximum Dimension ◽  
Point Transformations

Complex Lie point transformations are used to linearize a class of systems of second order ordinary differential equations (ODEs) which have Lie algebras of maximum dimensiond, withd≤4. We identify such a class by employing complex structure on the manifold that defines the geometry of differential equations. Furthermore we provide a geometrical construction of the procedure adopted that provides an analogue inR3of the linearizability criteria inR2.

scholarly journals Symmetry Classification of First Integrals for Scalar Linearizable Second-Order ODEs

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
K. S. Mahomed ◽  
E. Momoniat
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Complete Classification ◽  
First Integrals ◽  
Second Order ◽  
Point Transformations ◽  
Maximal Algebra ◽  
The Relationship ◽  
Linear Odes

Symmetries of the fundamental first integrals for scalar second-order ordinary differential equations (ODEs) which are linear or linearizable by point transformations have already been obtained. Firstly we show how one can determine the relationship between the symmetries and the first integrals of linear or linearizable scalar ODEs of order two. Secondly, a complete classification of point symmetries of first integrals of such linear ODEs is studied. As a consequence, we provide a counting theorem for the point symmetries of first integrals of scalar linearizable second-order ODEs. We show that there exists the 0-, 1-, 2-, or 3-point symmetry cases. It is shown that the maximal algebra case is unique.

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