Theorems related to the Sturm-Picone comparison theorem. I was reading in some course notes on differential equations last year from a URL I can no longer find, and there was a problem that has itched me because I always felt like there was an error in the exercise problem.
Suppose $p(t)>0$ and $p(t)$ is continuous on the interval $(0,\infty)$. Prove that every non-trivial solution of 
$$ty^{\prime\prime} + p(t) y = 0$$
has infinitely many roots.
I feel like the solution should follow immediately from the Sturm-Picone comparison theorems, or at least it can be seen from their proofs. But I've gone around in circles long enough to think this statement cannot be true, but I can't prove it one way or another.  

This is not true as stated. Let $p(t)=\frac{1}{4}t^{-1}$; then the equation $ty''+py=0$ has solution $y(t)=t^{1/2}$. 
