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So what I have been trying to do is solve it for ##t<t^{\prime}## in which case the boundary conditions at ##0## apply. Then solve it for ##t>t^{\prime}## (without the boundary conditions at 0). And apply a continuity condition at ##t^{\prime}## kind of like ##G(t^{\prime},t^{\prime})=G(t^{\prime},t^{\prime})##. I know I am supposed to be able to find all of the constants (in terms of ##t^{\prime}##), but there are not enough conditions on this solution to find all of them. What am I missing?

Thank you very much in advance! I would be happy to also ask in terms of a specific example but I'm afraid it would be removed.