Chapter 8.5, Problem 28P

The general solution of forced undamped spring-mass system s is

$y\left(t\right)=A_0\cos \left({\omega }_{0}t\right)+\frac{F_0}{m\left({\omega }_0{}^2-{\omega }^2\right)}\cos \left(\omega t\right)$

if $\frac{\omega }{{\omega }_0}=\frac{p}{q}$, where $p$ and $q$ are integers, show that motion is periodic with $T=\frac{2\pi q}{{\omega }_0}$