# Invariant characterization of scalar third-order ODEs that admit the maximal contact symmetry Lie algebra

## TL;DR

This paper uses Cartan's method to find invariants characterizing third-order ODEs with maximal contact symmetry, enabling efficient reduction to linear form and illustrating the approach with examples.

## Contribution

It provides an invariant characterization and auxiliary functions for third-order ODEs with maximal contact symmetry using Cartan's equivalence method.

## Key findings

- Derived invariant conditions for maximal contact symmetry
- Provided auxiliary functions for contact transformation
- Illustrated method with multiple examples

## Abstract

The Cartan equivalence method is utilized to deduce an invariant characterization of the scalar third-order ordinary differential equation $u'''=f(x,u,u',u'')$ which admits the maximal ten-dimensional contact symmetry Lie algebra. The method provides auxiliary functions which can be used to efficiently determine the contact transformation that does the reduction to the simplest linear equation $\bar{u}'''=0$. Furthermore, ample examples are given to illustrate our method.