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 Slinearizable 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.
An alternative proof of Lie's approach for linearization of scalar second order ODEs is derived using the relationship between λ-symmetries and first integrals.This relation further leads to a new λ-symmetry linearization criteria for second order ODEs which provides a new approach for constructing the linearization transformations with lower complexity. The effectiveness of the approach is illustrated by obtaining the local linearization transformations for the linearizable nonlinear ODEs of the form y ′′ + F 1 (x, y)y ′ + F (x, y) = 0. Examples of linearizing nonlinear ODEs which are quadratic or cubic in the first derivative are also presented.
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