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.