Laplace II: Steps, Impulses & Convolution · Superchallenge problem

An undamped oscillator at rest is struck with a unit impulse at $t=0,2\pi,4\pi,\ldots$: $y''+y=\displaystyle\sum_{k=0}^{\infty}\delta(t-2k\pi)$, $y(0^-)=y'(0^-)=0$. Find $y\!\left(2025\pi+\dfrac{\pi}{2}\right)$.