Why does Abel's identity imply either $W = 0$ or $W \neq 0$ everywhere? Let $y_1$ and $y_2$ be solutions to the linear differential equation $A(x)y'' + B(x)y' + C(x)y = 0 $ and let $W = W(y_1, y_2)$ be the Wronskian of the solutions. Why does Abel's identity 
$\displaystyle W(y_1, y_2)(x) = W(y_1, y_2)(x_0)\cdot exp\left(-\int{\frac{B(x)}{A(x)}dx}\right) $
imply the Wronskian is either zero everywhere or nonzero everywhere?
I know that $W \neq 0$ if the solutions are linearly independent and $W=0$ if the solutions are linearly dependent. But how does Abel's identity show this for all $x$ in the domain?
Thanks for the help.

Suppose that $W$ were ever zero. Since the $\exp$ term is never zero, that means the $W(y_1, y_2)(x_0)$ term must be zero. But that means that $W$ is zero everywhere.
