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.