Some Remarks on the Solution of Linearisable Second-Order Ordinary Differential Equations via Point Transformations

Abstract EN

Transformations of differential equations to other equivalent equations play a central role in many routines for solving intricate equations.

A class of differential equations that are particularly amenable to solution techniques based on such transformations is the class of linearisable second-order ordinary differential equations (ODEs).

There are various characterisations of such ODEs.

We exploit a particular characterisation and the expanded Lie group method to construct a generic solution for all linearisable second-order ODEs.

The general solution of any given equation from this class is then easily obtainable from the generic solution through a point transformation constructed using only two suitably chosen symmetries of the equation.

We illustrate the approach with three examples.