Green's Function using Laplace Transformation

I was wondering if someone could help me go through a simple example in using Green's Function.

Lets say:
x' + x = f(t)
with an initial condition of x(t=0,t')=0;

Step 1 would be to re-write this as:
G(t,t') + G(t,t') = $\delta$(t-t')

then do you multiply by f(t')$\oint$dt' ?
which I would believe would give me:

s G(s) + G(s) = e^-st

and G(s) = $\frac{1}{s+1}$ e^-st'
then giving me my G(t,t') = e^-(t-t') * U(t-t') ?

Not sure if that is the expected Green's function or if I screwed up somewhere.

Also, if f(t) = U(t-1), what would be the system's response?
* U fxn is a Heaviside step function