Neumann Green function for a second-order ordinary differential equation

Let $y_1,y_2$ solve $y^{\prime \prime }+\alpha y^{\prime }+\beta y=0$ with $y_1^{\prime }(0)=0$ and $y_2^{\prime }(1)=0$. For Neumann boundary conditions, the Green function iswhere $W=y_1{y}_2^{\prime }-y_1^{\prime }{y}_2$. The derivative jump is one. If $\alpha =0$, the Abel identity makes $W$ constant and $G$ symmetric.

For real $\kappa \ne 0$, this Green function inverts $d^2/d{x}^2-{\kappa }^2$ on $[0,1]$ with homogeneous Neumann boundary conditions. It is continuous and its first derivative has jump one at $x=\xi$. At $\kappa =0$, a Green function with the same unit Dirac delta source and zero boundary derivatives does not exist: integrating its defining equation would give zero equal to one. The equation $y^{\prime \prime }=f$ with these boundary conditions instead requires ${\int }_0^{1}f(x)dx=0$ and determines $y$ only up to a constant.