  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.
