Problem Prove that if the Wronskian of any two solutions of differential equation $y''+p(x)y'+q(x)y=0$ is constant, then $p(x)$ is zero.
My attempt. : Let $y_1$ and $y_2$ be two solutions of given differential equation. Note that the Wronskian $W=W[y_1,y_2]$ satisfies $W'+p(x)W=0$. Since $W$ is constant, we get $p(x)W=0$.
Question. How do I show that $W$ cannot be equal to zero? If there are two solutions $y_1$, $y_2$ that are linearly independent, then $W[y_1,y_2]\neq 0$. But I am not certain of the existence of such solutions.
My question is : If a second-order ODE has a solution, do there exist two solutions that are linearly independent?