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

SIGMA 6 (2010), 051, 11 pages      arXiv:1006.2891

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

Warisa Nakpim and Sergey V. Meleshko
School of Mathematics, Institute of Science, Suranaree University of Technology, Nakhon Ratchasima, 30000, Thailand

Received January 18, 2010, in final form June 03, 2010; Published online June 15, 2010

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.

Key words: linearization problem; generalized Sundman transformations; nonlinear second-order ordinary differential equations.

pdf (222 kb)   ps (135 kb)   tex (12 kb)


  1. Meleshko S.V., Methods for constructing exact solutions of partial differential equations, Mathematical and Analytical Techniques with Applications to Engineering, Springer, New York, 2005.
  2. Ibragimov N.H., Elementary Lie group analysis and ordinary differential equations, Wiley Series in Mathematical Methods in Practice, Vol. 4, John Wiley & Sons, Ltd., Chichester, 1999.
  3. Lie S., Classifikation und Integration von gewöhnlichen Differentialgleichungen zwischen x, y die eine Gruppe von Transformationen gestatten, Math. Ann. 32 (1888), 213-281.
  4. Duarte L.G.S., Moreira I.C., Santos F.C., Linearization under non-point transformations, J. Phys. A: Math. Gen. 27 (1994), L739-L743.
  5. Euler N., Wolf T., Leach P.G.L., Euler M., Linearisable third-order ordinary differential equations and generalised Sundman transformations: the case X'''=0, Acta Appl. Math. 76 (2003), 89-115, nlin.SI/0203028.
  6. Nakpim W., Meleshko S.V., Linearization of third-order ordinary differential equations by generalized Sundman transformations: the case X'''+αX=0, Commun. Nonlinear Sci. Numer. Simul. 15 (2010), 1717-1723.
  7. Berkovich L.M., The integration of ordinary differential equations: factorization and transformations, Math. Comput. Simulation 57 (2001), 175-195.
  8. Berkovich L.M., Factorization and transformations of differential equations. Methods and applications, R&C Dynamics, Moscow, 2002 (in Russian).
  9. Karasu A., Leach P.G.L., Nonlocal symmetries and integrable ordinary differential equations: x''+3xx'+x3=0 and its generalizations, J. Math. Phys. 50 (2009), 073509, 17 pages.

Previous article   Next article   Contents of Volume 6 (2010)