Lie group symmetries for first order ODEs

TL;DR

This paper investigates the use of Lie group symmetries to solve first order ODEs, challenging previous assumptions about the difficulty of finding symmetry solutions and exploring geometric intuition for simplifying the process.

Contribution

It introduces methods to bypass the complex linearized symmetry condition using inspired guesswork and geometric intuition, enhancing the symmetry-based solution approach for first order ODEs.

Findings

Connection between prolongation and linearized symmetry condition verified

Symmetry solutions can be found without solving the full linearized condition

Geometric intuition aids in identifying symmetry solutions

Abstract

This paper is centred on solving differential equations by symmetry groups for first order ODEs and is in response to Starrett (2007). It also explores the possibility of averting the assumptions by Olver (2000) that, in practice finding the solutions of the linearized symmetry condition is usually a much more difficult problem than solving the original ODE but, by inspired guesswork or geometric intuition, it is possible to ascertain a particular solution of the linearized symmetry condition. In addition, there is the verification of the connection between prolongation and the linearized symmetry condition as used in solving first order ODEs.