Nonhomogeneous Equations & Resonance · Superchallenge problem

For $\omega\ne1$ let $y_\omega$ solve $y''+y=\cos\omega t$, $y(0)=0$, $y'(0)=0$. Compute $\displaystyle\lim_{\omega\to1}y_\omega\!\left(\dfrac{\pi}{2}\right)$.