Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 041, 10 pages      arXiv:1302.3242      https://doi.org/10.3842/SIGMA.2013.041

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

M. Tahir Mustafa, Ahmad Y. Al-Dweik and Raed A. Mara'beh
Department of Mathematics & Statistics, King University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

Received February 16, 2013, in final form May 25, 2013; Published online May 31, 2013

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.

Key words: linearization problem; generalized Sundman transformations; first integrals; nonlinear second-order ODEs.

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