Let $y_1$, $y_2$ be two solutions of a homogeneous linear second order differential equation $y''$ + $p(t)y'$ + $q(t)y$ = $0$ over the interval $\alpha$ < $t$ < $\beta$.
Prove that the wronskian $W[y_1, y_2]$ := $y_1y_2'$ - $y_1'y_2$ is zero or nonzero over the whole interval $\alpha$ < $t$ < $\beta$, that is, if $W$ is zero(respectiely nonzero) at one point $t_0$ in the interval $\alpha$ < $t$ < $\beta$, then $W$ should be zero(respectively nonzero) on all points in the interval.
I saw this problem on my book, and I have no idea how to approach. I know that $W[y_1, y_2] = 0$ means that $y_1$ and $y_2$ are linearly dependent, and I think we should use this fact, but how? Can anyone help me?
Hint
$$ W'= y_1'y_2'+y_1y_2''-y_1''y_2-y_1'y_2'=y_1y_2''-y_1''y_2 \\ =y_1(-p(t)y_2'-q(t)y_2)-y_2(-p(t)y_1'-q(t)y_1)=-p(t)W $$