A systematic method of finding linearizing transformations for nonlinear ordinary differential equations: I. Scalar case

TL;DR

This paper introduces a systematic, efficient method to identify the maximum number of linearizing transformations for nonlinear ODEs, including new types, using fewer integrals of motion, applicable to scalar and coupled systems.

Contribution

The paper presents a novel algorithm that unearths multiple linearizing transformations for nonlinear ODEs, expanding beyond known types and applicable to scalar and coupled equations.

Findings

Uncovered a new type of linearizing transformation in third order ODEs.

Developed an algorithm to derive maximal linearizing transformations.

Demonstrated the method with multiple examples for scalar and coupled ODEs.

Abstract

In this set of papers 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 focuss 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.