Prove or give a counterexample that any non-trivial solution of $y^{\prime\prime} + r(x)y = 0$ on a finite interval has at most finite number of zeros.
If the finite interval is closed, then we can assume that there are infinitely many zeroes and argue that this set of zeroes must have a limit point hence arriving at a contradiction.
Also, if we assume $r(x)$ to be bounded (by $M$), then we can use Sturm Comparison against $y^{\prime\prime} + My = 0$ to arrive at a contradiction. So, if there is a counterexample, then $r$ must be unbounded.
But, I can't proceed anymore.