Linearization of Second-Order Ordinary Differential Equations by Generalized Sundman Transformations

TL;DR

This paper examines the limitations of previous methods for linearizing second-order ordinary differential equations using generalized Sundman transformations, demonstrating that earlier solutions are incomplete and introducing new insights into the linearization process.

Contribution

The paper reveals the incompleteness of prior solutions and shows that the Laguerre form is insufficient for linearization via generalized Sundman transformations.

Findings

Previous solutions are incomplete

Laguerre form is not sufficient for linearization

New examples demonstrate limitations of earlier methods

Abstract

The linearization problem of a second-order ordinary differential equation by the generalized Sundman transformation was considered earlier by Duarte, Moreira and Santos using the Laguerre form. The results obtained in the present paper demonstrate that their solution of the linearization problem for a second-order ordinary differential equation via the generalized Sundman transformation is not complete. We also give examples which show that the Laguerre form is not sufficient for the linearization problem via the generalized Sundman transformation.