Wronskian of two independent solutions equaling zero at a specific point only? Given $y_1(x)=\sin(x^2)$ and $y_2(x)=\cos(x^2)$, I constructed a linear, homogenic ODE of order 2 by solving:
$$
        \begin{vmatrix}
        y & y_1 & y_2 \\
        y' & y_1' & y_2' \\
        y'' & y_1'' & y_2'' \\
        \end{vmatrix}=0
$$
Now, I noticed that the Wronskian of $y_1$ and $y_2$ at $x=0$ equals $0$. But the Wronskian of independent solutions is never $0$. And if it $0$ at one point, it is zero everywhere, which I don't see happening here, as
$$
        W(y_1,y_2)=\begin{vmatrix}
         y_1 & y_2 \\
        y_1' & y_2' \\
       \end{vmatrix}=-2x
$$
How does this make sense?

make sure that the coefficient of the highest order derivative doesn't vanish anywhere in the domain and the coefficient of the rest of the derivatives are continuous.specially check at x=0
