International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 59619, 25 pages
Research Article

Use of a Strongly Nonlinear Gambier Equation for the Construction of Exact Closed Form Solutions of Nonlinear ODEs

M. P. Markakis

Department of Engineering Sciences, University of Patras, Patras 26500, Greece

Received 21 March 2007; Revised 29 May 2007; Accepted 5 June 2007

Academic Editor: Martin J. Bohner

Copyright © 2007 M. P. Markakis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We establish an analytical method leading to a more general form of the exact solution of a nonlinear ODE of the second order due to Gambier. The treatment is based on the introduction and determination of a new function, by means of which the solution of the original equation is expressed. This treatment is applied to another nonlinear equation, subjected to the same general class as that of Gambier, by constructing step by step an appropriate analytical technique. The developed procedure yields a general exact closed form solution of this equation, valid for specific values of the parameters involved and containing two arbitrary (free) parameters evaluated by the relevant initial conditions. We finally verify this technique by applying it to two specific sets of parameter values of the equation under consideration.