Let $q$ be a real valued non-trivial solution solution of $ y'' +A(x)y = 0 \text{ on } a
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 $(x_1, x_2)$