If $$y_1=\sin(x^2)$$ $$y_2=\cos(x^2)$$ are linearly independent solutions of $$xy''-y'+4x^3y=0$$ the wronskian $$W(y_1,y_2)$$ is zero when $x=0$.
Does this contradict the theorem which states that if y1 and y2 are linearly independent solution then their wronskian is not zero ?
Are $y_1$ and $y_2$ still independent when x=0 but the wronskian is zero ?
Are $y_1$ and $y_2$ independent for all values of $x$ but dependent when $x=0$ ?
Is this related to the uniqueness theorem ? how ?