International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 59619, 25 pages
doi:10.1155/2007/59619
Research Article
Use of a Strongly Nonlinear Gambier Equation for the Construction of Exact Closed Form Solutions of Nonlinear ODEs
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.
Abstract
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.