Reduction of Fifth-Order Ordinary Differential Equations to Linearizable form by Contact Transformations
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Reduction of Fifth-Order Ordinary Differential Equations to Linearizable form by Contact Transformations Supaporn Suksern1,2
© Foundation for Scientific Research and Technological Innovation 2017
Abstract This paper is devoted to the study of the linearization problem of fifth-order ordinary differential equations by means of contact transformations. The necessary and sufficient conditions for linearization are obtained. The procedure for obtaining the linearizing transformations is provided in explicit form. Examples demonstrating the procedure of using the linearization theorems are presented. Keywords Linearization problem · Point transformation · Contact transformation · Nonlinear ordinary differential equation
Introduction Nonlinear differential equations are used to describe a great variety of phenomena in almost every branch of sciences. In general, however, it is evidently difficult to solve them directly. One of a powerful methods is to transform them into the linear differential equations, which is called linearization. The problem of linearization of ordinary differential equations attracted attention of many mathematicians such as S. Lie and E. Cartan. In 1883, the first linearization problem for ordinary differential equations was solved by Lie [1]. He found the general form of all ordinary differential equations of second order that can be reduced to a linear equation by changing the independent and dependent variables. He showed that any linearizable secondorder equation should be at most cubic in the first-order derivative and provided a linearization test in terms of its coefficients . The linearization criterion is written through relative invariants
This research was partially supported by Naresuan University.
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Supaporn Suksern [email protected]
1
Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
2
Research center for Academic Excellence in Mathematics, Naresuan University, Phitsanulok 65000, Thailand
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Differ Equ Dyn Syst
of the equivalence group. Liouville [2] and Tresse [3] treated the equivalence problem for second-order ordinary differential equations in terms of relative invariants of the equivalence group of point transformations. Lie also noted that all second-order ordinary differential equations can be mapped to each other by means of contact transformations, and that this is not so for third-order ordinary differential equations. Hence, the linearization problem using contact transformations becomes interesting for ordinary differential equations of order greater than two. There are other approaches for solving the linearization problem of a second-order ordinary differential equation. For example, one was developed by Cartan [4]. The idea of his approach was to associate with every differential equation, a uniquely defined geometric structure of a certain form. Another approach makes use of the generalized Sundman transformation [5]. Cartans approach was further applied by Chern [6] to third-order ordinary differential e
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