On the Linearization of Second-Order Ordinary Differential Equations to the Laguerre Form via Generalized Sundman Transformations

TL;DR

This paper presents a new criterion for linearizing nonlinear second-order ODEs to the Laguerre form using generalized Sundman transformations, enabling explicit solutions and applications to geodesics on surfaces of revolution.

Contribution

It introduces a novel characterization of S-linearizable equations based on ODE coefficients and an auxiliary function, simplifying the construction of linearizing transformations.

Findings

Derived explicit general solutions for first integrals.

Applied the method to solve geodesic equations on surfaces of revolution.

Demonstrated the approach's effectiveness with a unified solution procedure.

Abstract

The linearization problem for nonlinear second-order ODEs to the Laguerre form by means of generalized Sundman transformations (S-transformations) is considered, which has been investigated by Duarte et al. earlier. A characterization of these S-linearizable equations in terms of first integral and procedure for construction of linearizing S-transformations has been given recently by Muriel and Romero. Here we give a new characterization of S-linearizable equations in terms of the coefficients of ODE and one auxiliary function. This new criterion is used to obtain the general solutions for the first integral explicitly, providing a direct alternative procedure for constructing the first integrals and Sundman transformations. The effectiveness of this approach is demonstrated by applying it to find the general solution for geodesics on surfaces of revolution of constant curvature in a unified manner.