Anharmonic Oscillator

• A point particle of mass $m$, is bounded by an even potential

\[V(x) = \frac12m\omega_0^2 x^2 + \sum_{r=2}^\infty \lambda_r \frac{x^{2r}}{2r}\]

If its total energy is $E$, find a perturbative expression up to second order in {$\lambda_r$} for the frequency of the oscillation.

Hint:

\[\sin^{2r-1}(\theta) = \frac{-1}{4^{r-1}}\sum_{s=1}^r(-)^s\binom{2r-1}{r+s-1}\sin\big[(2s-1)\theta\big]\]