On the linearization of second-order ordinary differential equations

Abstract

In this Letter, the problem of characterizing all second-order ordinary differential equations y″=f(x, y, y′) which are locally linearizable by a change of dependent and independent variables (x, y)→(X, Y) is considered. Since all second-order linear equations are locally equivalent to y″=0, the problem amounts to finding necessary and sufficient conditions for y″=f(x, y, y′) to be locally equivalent to y″=0. It turns out that two apparently different criteria for linearizability have been formulated in the literature: the one found by M. Tresse and later rederived by É. Cartan, and the criterion recently given by Arnol'd [Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New York, 1983]. It is shown here that these two sets of linearizability conditions are actually equivalent. As a matter of fact, since Arnol'd's criterion is stated without proof in the latter reference, this work can be alternatively considered as a proof of Arnol'd's linearizability conditions based on Cartan-Tresse's. Some further points in connection with the relationship between Arnol'd's and Cartan-Tresse's treatment of the linearization problem are also discussed and illustrated with several examples.