Integrability cases for the anharmonic oscillator equation

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 $\frac{d^{2}x}{d{t}^2}+f_1\left(t\right)\frac{dx}{dt}+f_2\left(t\right)x+f_3\left(t\right)x^n=0$. The first exact solution is obtained from a particular solution of the point transformed equation $d^{2}X/d{T}^2+X^n\left(T\right)=0$, $n\notin \left\{-3,-1,0,1\right\}$, 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 $\frac{1}{f_3(t)}\frac{d{f}_3}{dt}$ respectively. By reducing the integrability condition to a Bernoulli type equation, two exact classes of solutions of the anharmonic oscillator equation are obtained.

7 pages, no figures

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