I am trying to prove a regular SL-BVP is self adjoint.
I'm using the general case: $\frac{d}{dx}[p(x)y'(x)]+q(x)y(x)=f(x)$ with boundry $a_1y(a)+a_2y'(a)=0$ and $b_1y(b)+b_2y'(b)=0$
So far I've tried using the form $A_2(x)y''(x)+A_1y'(x)+A_0y(x)$ the calculating the formal adjoint
$L^+[y](x)=[A_2(x)y(x)]'' - [A_1(x)y(x)]'+A_0y(x)$
I ran through this and got $A_2(x)y''(x)+(2A_2'(x)-A_1(x))y'(x)+(A_2''(x)-A_1'(x)+A_0(x))y(x)$
If $A_2$ is only twice differentiable and $A_1$ is only once differentiable I can kind of start seeing the right outcome but it is not quite it.
Maybe I am way off base with how to even start this problem, I can kind of see the right outcome but it is always slightly off. Any hints?
\begin{align} \int [py_1']'y_2 dx &= py_1'y_2 - \int py_1'y_2' dx \\ &= py_1'y_2 - \int y_1'[py_2']dx \\ &= py_1'y_2 -y_1[py_2']+\int y_1[py_2']'dx \end{align}