T. Harko, F. S. N. Lobo, M. K. Mak

Abstract
Using N. Euler's theorem on the integrability of the general anharmonic oscillator equation cite{12}, we present three distinct classes of general solutions of the highly nonlinear second order ordinary differential equation $frac{d^{2}x}{dt^{2}}+f_{1}left(t ight) frac{dx}{dt}+f_{2}left(t ight) x+f_{3}left(t ight) x^{n}=0$. The first exact solution is obtained from a particular solution of the point transformed equation $d^{2}X/dT^{2}+X^{n}left(T ight) =0$, $n otin left{-3,-1,0,1 ight} $, which is equivalent to the anharmonic oscillator equation if the coefficients $f_{i}(t)$, $i=1,2,3$ satisfy an integrability condition. The integrability condition can be formulated as a Riccati equation for $f_{1}(t)$ and $frac{1}{f_{3}(t)}frac{df_{3}}{dt}$ respectively. By reducing the integrability condition to a Bernoulli type equation, two exact classes of solutions of the anharmonic oscillator equation are obtained.

Keywords
Mathematical: Physics - Nonlinear: Sciences - Exactly: Solvable: and: Integrable: Systems

Journal of Pure and Applied Mathematics: Advances and Applications
Volume 10, Number 1, Page 1_115
2013 April

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