Final Exam Preparation Frustration

Hi,

I am one exam away from finishing my undergraduate degree (Rock) which is in a few days, but I am struggling with a particular concept which is Green's functions for PDE's. I really struggle with this for some reason. I have a number of questions that I will post with my attempt at a solution. Here is my first question:

Let be a bounded region in . Let be a Green's function such that

Show that the solution of the equation

is

Here is my attempt at a solution....but I am sure if my working is sufficient....

(this is one of Green's Identities)

therefore