Hint on how to solve a Boundary value problem with Green's function

Question

Can someone give me hints/directions how to figure this problem out? Thanks

$\frac{d^2G}{dx^2} = \delta(x-\bar{x})$

$G(0) = G(L) = 0$

Answer 1

It's obvious that $G(x)$ is continuous.

Let $G(\bar x)=g$. We can solve two boundary value problems

$G''(x)=0, G(0)=0, G(\bar x)=g$

and

$G''(x)=0, G(\bar x)=g, G(L)=0$

and then find such $g$ that $G'(\bar x+0)-G'(\bar x-0)=1$.