# 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?

• I agree. Indeed, at $x=0$ the coefficient of $y''$ is the determinant $$\det\begin{bmatrix} 0 & 1 \\ 0 & 0\end{bmatrix}$$ Feb 8 '16 at 21:13