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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