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

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.

A Systematic Method of Finding Linearizing Transformations for Nonlinear Ordinary Differential Equations II: Extension to Coupled ODEs

“…Once this procedure is realized, the remaining part will be in identifying linearizing point transformations from the given two integrals. However, this procedure has already been worked out in Part-I (for the scalar case) [1]. The result reveals that one can identify the following three pairs of point transformations from a given integral, namely…”

“…where a (1,2) , b (1,2) , c (1,2) , d (1,2) and e (1,2) are arbitrary real numbers. Here K (i) (1) and K (i)…”

“…We have seen in Part-I [1] that one can derive three sets of point transformations and infinite number of ST/GLT from a given integral. Similar results hold good here also.…”

“…In the previous paper we have confined our studies to scalar nonlinear ODEs only. In the present paper we extend the method described in Part-I [1] to the case of two coupled second order nonlinear ODEs. Our results show that in such coupled ODEs there exists a wider class of linearizing transformations including hybrid ones (for example point-Sundman transformation, point-generalized linearizing transformation and Sundman-generalized linearizing transformation).…”

Periodic Solutions of a Class of Second-Order Differential Equation

“…In a recent paper [3] (see also [4] [5]), the second-order differential Equation (1) has been studied when 0 F H = = . A study of coupled quadratic unharmonic oscillators in terms of the Painlevé analysis and inte-grability can be seen in [6], and studies on the second-order differential equations can be seen in [7].…”