Let $y_{1}$ and $y_{1}$ be solutions of $y''+p(x)\ y'+q(x)\ y = r(x)$ on $[a,b]$, where $P,Q,R$ are continuous functions on $[a,b]$. Prove that either $y_{1} =y_{2}$ or $\{ x\in [a,b]\ :\ y_{1}(x) = y_{2}(x)\}$ is finite.
It is known that there is a unique solution, so if $y_{1},y_{2}$ are both solutions, then by the Wronskian, $\frac{y'_{1}}{y_{1}}=\frac{y'_{2}}{y_{2}}$ or $\frac{y'_{1}}{y_{1}}\not=\frac{y'_{2}}{y_{2}}$. But how do proceed after this? Help!!
$u=y_2-y_1$ solves $u''+pu'+q=0$. If the claim were false $u$ could be a non-zero function with infinitely many roots in $[a,b]$.
What do you know about infinite subsets in bounded intervals? What consequence does the limit point have for the derivative $u'$ in that point and thus the solution $u$, invoking uniqueness?