[Math] Does vanishing of wronskian of solutions at point $\implies$ solutions are linearly dependent

Question

Let $u$ and $v$ be two solutions of $y''+P(x)y'+Q(x)y=0$,Let $W(u,v)$ denote the wronskian of $u$ and $v$ then $W(u,v)$ vanishes at a point $x_0\in[a,b]\implies u$ and $v$ are linearly dependent

$W(u,v)(x_0)=0 $ for some $x_0\in [a,b]
\implies W(x)=0~,~~ \forall x\implies W(x)$ is identically zero on $[a,b]\implies u$ and $v$ are linearly dependent

Where I'm commiting mistake?

Answer 1

Hint:

The theorem is:

If the Wronskian $W(u,v)(x_0)$ is nonzero for some $x_0 \in [a,b]$ then $u$ and $v$ are linearly independent on $[a,b]$.

The contraposition is:

If $u$ and $v$ are linearly dependent then the Wronskian is zero for all $x \in [a,b]$.