On Computing Linearizing Coordinates From Symmetry Algebra

TL;DR

This paper characterizes the symmetry algebra of certain linearizable third order ODEs, providing a method to determine unique linearizing coordinates from the algebra, with practical examples.

Contribution

It introduces an algebraic characterization and an algorithm to find linearizing coordinates for third order linearizable ODEs based on their symmetry algebra.

Findings

Symmetry algebra uniquely determines the linearizing coordinates.

An explicit algorithm for finding linearizing coordinates is provided.

Illustrative examples demonstrate the procedure's effectiveness.

Abstract

A characterization of the symmetry algebra of the $n$th order ordinary differential equations (ODEs) with maximal symmetry and all third order linearizable ODEs is given. This is used to show that such an algebra $\mathfrak{g}$ determines $-$ up to a point transformation $-$ only one linear equation whose symmetry algebra is $\mathfrak{g}$ and an algorithmic procedure is given to find the linearizing coordinates. The procedure is illustrated by several examples from literature.