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

Abstract

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.

Linearizability of Nonlinear Third-Order Ordinary Differential Equations by Using a Generalized Linearizing Transformation

“…We begin with investigating the necessary conditions for linearization, that is, the general form of third-order equation (4) that can be obtained from a linear equation (6) by any generalized linearizing transformation (5). Applying a generalized linearizing transformation (5), one obtains the following transformation of the third-order derivatives:…”

“…Theorem 1. Any third-order ordinary differential equation (4) obtained from a linear equation (6) by a generalized linearizing transformation (5) has to be in the form (8).…”

“…A characterization of these equations that can be linearized by means of generalized Sundman transformations in terms of first integral and procedure for construction of linearizing transformations has been given by Muriel and Romero [4]. In [5], Mustafa et al gave a new characterization of linearizable equations in terms of the coefficients of ordinary differential equations and one auxiliary function. In [6], Nakpim and Meleshko pointed out that the solution of the linearization problem for a second-order ordinary differential equation via the generalized Sundman transformation considered earlier by Duarte et al [3] using the Laguerre form is not complete.…”

“…Sufficient conditions for equation(8) to be linearizable via the generalized linearizing transformation(5) with ̸ = 0 are as follows. (a) If 1 ̸ = 0, then the conditions are (18), (40), (41), and (42).…”

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

“…In addition, Muriel and Romero [21] characterized the equations that can be linearized by means of generalized Sundman transformations in terms of first integral. A new characterization of linearizable equations in terms of the coefficients of ordinary differential equation and one auxiliary function was given by Mustafa et al [22]. Moreover, Nakpim and Meleshko [23] pointed out that the solution given by Duarte et al using the Laguerre form is not complete.…”