`Journal of Applied MathematicsVolume 2012 (2012), Article ID 286290, 16 pageshttp://dx.doi.org/10.1155/2012/286290`
Research Article

## Analytical Approximate Solutions for the Cubic-Quintic Duffing Oscillator in Terms of Elementary Functions

1Departamento de Física, Ingeniería de Sistemas y Teoría de la Señal, Universidad de Alicante, Apartado 99, 03080 Alicante, Spain
2Instituto Universitario de Física Aplicada a las Ciencias y las Tecnologías, Universidad de Alicante, Apartado 99, 03080 Alicante, Spain
3Departamento de Ciencias Médicas, Facultad de Medicina, Universidad de Castilla-La Mancha, C/Almansa No. 14, 02006 Albacete, Spain
4Departamento de Física Aplicada, Escuela Superior de Ingeniería Informática, Universidad de Castilla-La Mancha, Avenida de España s/n, 02071 Albacete, Spain

Received 28 June 2012; Accepted 13 August 2012

Academic Editor: Livija Cveticanin

Copyright © 2012 A. Beléndez et al. 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

Accurate approximate closed-form solutions for the cubic-quintic Duffing oscillator are obtained in terms of elementary functions. To do this, we use the previous results obtained using a cubication method in which the restoring force is expanded in Chebyshev polynomials and the original nonlinear differential equation is approximated by a cubic Duffing equation. Explicit approximate solutions are then expressed as a function of the complete elliptic integral of the first kind and the Jacobi elliptic function cn. Then we obtain other approximate expressions for these solutions, which are expressed in terms of elementary functions. To do this, the relationship between the complete elliptic integral of the first kind and the arithmetic-geometric mean is used and the rational harmonic balance method is applied to obtain the periodic solution of the original nonlinear oscillator.