Classification of Scalar Fourth Order Ordinary Differential Equations Linearizable via Generalized Lie–Bäcklund Transformations

Abstract

Lie had shown that there is a unique class of scalar second order ordinary differential equations (ODEs) that can be converted to linear form by point transformations. Mahomed and Leach had shown that for higher order (than 2) scalar ODEs there are always three classes. Separately, Chern had linearized two classes of third order ODEs by using contact transformations. We provided an (inclusive) classification for third order ODEs by using a generalization of contact transformations. Here we extend that work to the fourth order using a generalization of the Lie–Backlund transformation and demonstrate that there are (at least) four classes of fourth order linearizable ODEs.