Neumann Green function for a second-order ordinary differential equation
= Neumann Green function for a second-order ordinary differential equation
{c}
Let $y_1,y_2$ solve $y''+\alpha y'+\beta y=0$ with $y_1'(0)=0$ and $y_2'(1)=0$. For Neumann boundary conditions, the Green function is
$$
G(x,\xi)=\frac1{W(\xi)}
\begin{cases}
y_1(x)y_2(\xi),&x<\xi,\\
y_2(x)y_1(\xi),&x>\xi,
\end{cases}
$$
where $W=y_1y_2'-y_1'y_2$. The derivative jump is one. If $\alpha=0$, the Abel identity makes $W$ constant and $G$ symmetric.