# Some remarks on the solution of linearisable second-order ordinary differential equations via point transformations

## TL;DR

This paper presents a method using Lie group theory to construct a generic solution for all linearisable second-order ODEs, simplifying their solution process through point transformations and symmetry analysis.

## Contribution

It introduces a unified approach leveraging a specific characterization and expanded Lie group method to solve all linearisable second-order ODEs via point transformations.

## Key findings

- A generic solution framework for linearisable second-order ODEs.
- Simplification of solving such ODEs using symmetry-based point transformations.
- Validated approach with three illustrative examples.

## Abstract

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.