I was solving a question where I was asked to prove that if $p(x) = 0$ for $L(y) = y'' + p(x)y' + q(x)y = 0$ then the wronskian of the two independent solutions is a constant.
I was able to prove that $W(y_1, y_2)(x) = W(y_1, y_2)(x_0)$ for all $x$. I feel like this is a constant but wasn't sure if it's okay to draw that conclusion. Like am I missing a piece?
Any help would be appreciated. Thanks!
On
If you have shown that $W[y_1,y_2](x)=W[y_1,y_2](x_0)$ for all $x>x_0$ then you have shown constancy.
I believe that an easy way to do this is to use Abels' Identity (basic proof here)
This is that $W[y_1,y_2](x)=W[y_1,y_2](x_0)\cdot e^{-\int\limits_{x_0}^{x}p(\xi)d\xi}$
Thus if $p(\xi)=0$, the above equation reduces to $W[y_1,y_2](x)=W[y_1,y_2](x_0)\cdot e^{0}=W[y_1,y_2](x_0)$
I think this should answer your question.
\begin{align} W &= y_1'y_2 - y_1 y_2' \\ W' &= y_1''y_2 - y_1 y_2'' \\ W' &= -q y_1 y_2 + q y_1y_2 = 0. \end{align}