A Systematic Method of Finding Linearizing Transformations for Nonlinear Ordinary Differential Equations I: Scalar Case

DOI

10.1142/S140292511250012XHow to use a DOI?

Keywords

Nonlinear ODEs; linearization; first integrals

Abstract

In this paper we formulate a stand alone method to derive maximal number of linearizing transformations for nonlinear ordinary differential equations (ODEs) of any order including coupled ones from a knowledge of fewer number of integrals of motion. The proposed algorithm is simple, straightforward and efficient and helps to unearth several new types of linearizing transformations besides the known ones in the literature. To make our studies systematic we divide our analysis into two parts. In the first part we confine our investigations to the scalar ODEs and in the second part we focus our attention on a system of two coupled second-order ODEs. In the case of scalar ODEs, we consider second and third-order nonlinear ODEs in detail and discuss the method of deriving maximal number of linearizing transformations irrespective of whether it is local or nonlocal type and illustrate the underlying theory with suitable examples. As a by-product of this investigation we unearth a new type of linearizing transformation in third-order nonlinear ODEs. Finally the study is extended to the case of general scalar ODEs. We then move on to the study of two coupled second-order nonlinear ODEs in the next part and show that the algorithm brings out a wide variety of linearization transformations. The extraction of maximal number of linearizing transformations in every case is illustrated with suitable examples.

Copyright

© 2012 The Authors. Published by Atlantis Press and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - V. K. Chandrasekar
AU  - M. Senthilvelan
AU  - M. Lakshmanan
PY  - 2012
DA  - 2012/07/04
TI  - A Systematic Method of Finding Linearizing Transformations for Nonlinear Ordinary Differential Equations I: Scalar Case
JO  - Journal of Nonlinear Mathematical Physics
SP  - 182
EP  - 202
VL  - 19
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.1142/S140292511250012X
DO  - 10.1142/S140292511250012X
ID  - Chandrasekar2012
ER  -