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Let $q$ be a real valued non-trivial solution solution of $$ y'' +A(x)y = 0 \text{ on } a$w$ be a real valued non-trivial solution of $$ y'' + B(x)y = 0 \text{ on } a$A$ and $B$ are real valued continuous functions satisfying $$ B(x)>A(x) \text{ for } a$x_1$ and $x_2$ are successive zeros of $q$ on $(a,b)$, then $w$ must vanish at some point $p \in (x_1, x_2)$?

Partial answer: Let $q, w>0$ on $(x_1, x_2)$,then with $(wq'-qw')'= (B-A)qw$, and by integration from $x_1$ to $x_2$ we get $w(x_2)q'(x_2)-w(x_1)q'(x_1)> 0$. Somehow I want to show that that $q'(x_1)< 0$ or $q'(x_2)>0$, which will then contradict $q > 0$ on $(x_1, x_2)$

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