Find the exact solution of the nonlinear pendulum equation [ddot{theta}+omega_{0}^{2}left(theta-frac{theta^{3}}{6} ight)=0] with (dot{theta}=0) when (theta=theta_{0}), where (theta_{0})

Find the exact solution of the nonlinear pendulum equation

\[\ddot{\theta}+\omega_{0}^{2}\left(\theta-\frac{\theta^{3}}{6}\right)=0\]

with \(\dot{\theta}=0\) when \(\theta=\theta_{0}\), where \(\theta_{0}\) denotes the maximum angular displacement.