How does one use Green's function of the operator to get the solution of the arbitrary boundary value problem?

Question

Assume I've been given an operator $L$ and its Green's function $G(s, s')$. This is the function that solves the following: $$ L G(s, s') = \delta(s-s'), G(a,s') = G(b, s') = 0 $$ I know how to get a solution of the: $$ L u = f, u(a) = u(b) = 0 $$ But if the task was: $$ L u = f , u'(a) + D u(a) = A, u'(b) + C u(b) = B $$ Is there a way to even represent a solution with $G$? Or one would need to solve separately: $$ L u = 0 , u'(a) + D u(a) = A, u'(b) + C u(b) = B $$ and $$ L u = f , u'(a) + D u(a) = 0, u'(b) + C u(b) = 0 $$ Where the last one may be solved invoking new Green's function $N(s, s')$: $$ L N = 0 , N_s(a, s') + D N(a, s') = 0, N_s(b, s') + C N(b, s') = 0 $$