Chapter 8.3, Problem 38P

At time $t$ the displacement from equilibrium, $y\left(t\right)$, of an undamped spring-mass system of mass $m$ is governed by the initial value problem

$\frac{d^{2}y}{d{t}^2}+{\omega }^{2}y=\frac{F_0}{m}\cos \omega t,y\left(0\right)=1,y^{\prime }\left(0\right)=0$

Where $F_0$ and $\omega$ are positive constant. Solve this initial value problem to determine the motion of the system. What happens as $t\to \infty$?