For the ODE $$xy''-(x+2)y'+2y=0~,$$ which has solutions $y_1 = e^x$ and $y_2 = x^2+2x+2$, the Wronskian is $W=-e^x x^2$.
As per the known theorem, Wronskian is either identically zero (i.e. zero for all $x$) or is never zero.
But $-e^x x^2$ is zero at $x=0$, which appears to be in the domain of solutions of this equation (since $y(0) = y'(0)$.
Does this somehow contradict the theorem?