Consider the equation $$y''- \dfrac{y'}{x} = 0~.$$ Solution is $$y=Cx^2 + d~.$$ The Wronskian of $x^2$ and $1$ turns out to be $-2x$ which is zero at $x = 0$ and non zero elsewhere.
But the Wronskian of solutions to an equation of type $$y'' + p(x) y' + q(x) y = 0$$ should be identically zero or never zero.
What am I missing?