Classification of scalar third order ordinary differential equations linearizable via generalized contact transformations

TL;DR

This paper classifies scalar third order ODEs that can be linearized through generalized contact transformations, extending classical symmetry methods to higher orders and establishing new equivalence classes.

Contribution

It introduces a new type of transformations linking contact and point symmetries, leading to the classification of linearizable third order ODEs.

Findings

Four equivalence classes of linearizable third order ODEs identified

Relation established between contact symmetries and point symmetries

New transformations generalize classical linearization methods

Abstract

Whereas Lie had linearized scalar second order ordinary differential equations (ODEs) by point transformations and later Chern had extended to the third order by using contact transformation, till recently no work had been done for higher order or systems of ODEs. Lie had found a unique class defined by the number of infinitesimal symmetry generators but the more general ODEs were not so classified. Recently classifications of higher order and systems of ODEs were provided. In this paper we relate contact symmetries of scalar ODEs with point symmetries of reduced systems. We define new type of transformations that build up this relation and obtain equivalence classes of scalar third order ODEs linearizable via these transformations. Four equivalence classes of such equations are seen to exist.