Question

Let y=y(x) be a solution of (H). Suppose there is a number 'a' in I such that y(a)=y'(a)=0. Prove the p(x)=0 for all x in I. That is prove that y is the trivial solution. This is equivalent to proving that if y=y(x) is a nontrivial solution of (H), then y(x) and y'(x) can never be 0 simultaneously.