SIGMA 9 (2013), 041, 10 pages arXiv:1302.3242
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
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
linearization problem; generalized Sundman transformations; first integrals; nonlinear second-order ODEs.
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