ODE with Driac Delta Function

Studying for my PhD qualifying exam in PDE and came across a problem that I am unsure how to solve.

Let $\displaystyle t\in \mathbb{R}$ find a general solution to

$\displaystyle y'' - y = \delta(x)$.

So what I have so far is

$\displaystyle y'' - y = 0 \quad \forall t \ne \xi$

Thus, the general solution would be of the form

$\displaystyle y = A e^t + B e^{-t}$

We also know that the first derivative makes a negative one jump when $\displaystyle t = \xi$ so

$\displaystyle A_1 e^t - B_1 e^{-t} - A_2 e^t + B_2 e^{-t} = -1$

But this doesn't really seem to help me get to a final solution.

Any help would be greatly appreciated. Thank you in advance.