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