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Integrability cases for the anharmonic oscillator equation

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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Instituto de Astrofísica e Ciências do Espaço Universidade do Porto Faculdade de Ciências da Universidade de Lisboa Fundação para a Ciência e a Tecnologia
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