f" + 2af' + (aČ + bČ)f = g(x)
Boundary conditions: f(0) = f(1) = 0, and a and b are constants
Are you trying to find the Green's function for this problem?
The Green's function, G(x,y), for this problem must satify these conditions:
1) $\displaystyle G(x,y)"+ 2aG(x,y)'+ (a^2+ b^2)G(x,y)= 0$ for all $\displaystyle x\ne y$.
2) G(0,y)= G(1,y)= 0 for all y between 0 and 1.
3) $\displaystyle G'(x,y)^+- G'(x,y)- = -1$ where $\displaystyle G'(x,y)^+$ is the limit of G'(x,y) with respect to x as x approaches y from above and $\displaystyle G'(x,y)^-$ is the limit as x approaches y from below.
Find the solutions to (1) that satisfy G(0,y)= 0 and solutions that satisfy G(1,y)= 0. You will have two functions each with an unknown constant. Use the fact that they must match at x= y and (3) to determine those two constants.