Let . Then your equation is , which has solution . Plugging back in you get .
I started by looking for a Green's Function of the form
And thus we have two cases.
Applying our left-hand BC,
So to ensure that our solution is bounded at either extreme, we insist that is 0.
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
Which can be combined to give
All I'm really looking to do is make sure that I've correctly found the Green's function up to this point.