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.
A. Beléndez. M. L. Alvarez. J. Francés. S. Bleda. T. Beléndez. A. Nájera. E. Arribas. "Analytical Approximate Solutions for the Cubic-Quintic Duffing Oscillator in Terms of Elementary Functions." J. Appl. Math. 2012 (SI02) 1 - 16, 2012. https://doi.org/10.1155/2012/286290