Let . Then your equation is , which has solution . Plugging back in you get .
Subject to
I started by looking for a Green's Function of the form
For all
And thus we have two cases.
Case 1:
Implying that
Applying our left-hand BC,
So to ensure that our solution is bounded at either extreme, we insist that is 0.
Case 2:
Which also reduces to the form
Applying our right-hand BC,
So similarly, we insist that .
Combining our results, we have then that
However, we want to make sure that there's continuity at and a finite jump of 1 in the first derivative, so we get that
And
Which can be combined to give
and .
All I'm really looking to do is make sure that I've correctly found the Green's function up to this point.