I'm not looking for the complete answer of course, hopefully just a nudge in the right direction? I've tried part b) of this question for hours with little luck.
What I've done:
The equation can be rewritten as $$R''(r) + \frac{2}{r}R'(r) + k^2R(r) = J(r)$$ And $$L = D^2 + \frac{2}{r}D + k^2$$ where D is the derivative function. I also know that $$LG(r,s) = \delta(r-s)$$ for $0<r<∞$.
What I'm really unsure of is how to actually find G(r,s). I've tried the way suggested in my notes;
It didn't get me very far. I was unsure what to assign to l(r) and h(r) and all attempts have failed. Not even sure if the format for my trial of G(r,s) is correct? I'm probably missing something super easy but any help would be really appreciated.
HINT: To compute the Green's function from first principles (despite the exercise not asking you to do that) consider the function $u(r)=rG(r,s)$ and show that it satisfies the equation $$\frac{d^2u}{dr^2}+k^2u=\delta(r-s)$$
The Green's function computation should be relatively simple from there.