I am stuck on a possibly trivial question. I have the Greens function for the equation
$$p_0(x)y''+p_1(x)y'+p_2(x)y=0$$
with the boundary value $y(\alpha)=y(\beta)=0$ and I need to solve the equation
$$p_0(x)y''+p_1(x)y'+p_2(x)y=r(x)$$
with the boundary conditions $y(\alpha)=0, y(\beta) = A \not =0$
I know that if the boundary conditions of the second was the same as the first, then I could do
$$y(x)=\int_{\alpha}^{\beta} G(x,t) r(t) dt$$
But what does one do with the different boundary conditions?