On the linearization of some second-order ODEs via contact transformations

Only those second-order ordinary differential equations (ODEs) that admit an eight-dimensional Lie algebra of Lie point symmetries are linearizable via a point transformation. However, all second-order ODEs are linearizable via a contact transformation. Unfortunately, the latter result is not useful as one cannot usually find the contact symmetries of second-order ODEs. By analysing realizations of three-dimensional Lie algebras, we show how to explicitly construct the linearization of some second-order ODEs via contact transformations.