I solved it, I think. could be written as the composition of sines and cosines of (the monotonic function) when with a preceding (monotonic) factor , so should have infinitely many solutions in that case.
It turns out that makes the Wronskian vanish, so with a bit of effort, and can be found, which can be shown to be linearly independent solutions in this case. The general solution when , then, has finitely many zeros in .
I think this was somehow a stupid and overly lengthy approach to the problem, though, so suggestions for alternatives are welcome!