VTRMC 2006 Problem 4

We want to find functions $p(t)$ , $q(t)$ , $f(t)$ such that

(a) $p$ and $q$ are continuous functions on the open interval $(0,\pi)$ . (b) $f$ is an infinitely differentiable nonzero function on the whole real line $(-\infty,\infty)$ such that $f(0)=f'(0)=f''(0)$ . (c) $y=\sin t$ and $y=f(t)$ are solutions of the differential equation $y''+p(t)y'+q(t)y=0$ on $(0,\pi)$ .

Is this possible? Either prove this is not possible, or show this is possible by providing an explicit example of such $f,p,q$ .