Symmetry Classification of Scalar $n$th Order Ordinary Differential Equations

TL;DR

This paper completes the classification of scalar nth order ordinary differential equations based on their symmetry Lie algebras, revealing the possible symmetry structures and their limitations for orders n ≥ 4.

Contribution

It provides a complete symmetry classification for scalar nth order ODEs with n ≥ 4, identifying the maximum symmetry algebras and special cases like n=5.

Findings

n ≥ 4 equations do not admit an n+3 dimensional Lie algebra except for n=5

n ≥ 4 equations can have an n+2 dimensional Lie algebra leading to non-linearizable equations

For n ≥ 5, only one class of equations admits an n+2 dimensional Lie algebra

Abstract

We complete the Lie symmetry classification of scalar nth order, $n \geq 4$, ordinary differential equations by means of the symmetry Lie algebras they admit. It is known that there are three types of such equations depending upon the symmetry algebra they possess, viz. first-order equations which admit infinite dimensional Lie algebra of point symmetries, second-order equations possessing the maximum eight point symmetries and higher-order, $n \geq 3$, admitting the maximum $n + 4$ dimensional symmetry algebra. We show that nth order equations for $n \geq 4$ do not admit maximally an $n + 3$ dimensional Lie algebra except for $n = 5$ which can admit $sl(3, R)$ algebra. Also, they can possess an $n + 2$ dimensional Lie algebra that gives rise to a nonlinear equation that is not linearizable via a point transformation. It is shown that for $n \geq 5$ there is only one such class of equations.