$$\frac{dG}{dx} + G = \delta(x-x_0)\text{ with }G(x;x_0) = 0 $$
I am somewhat lost in what to do to turn this into a Green's function. My Prof went about it by simply using an integrating factor, but the reasoning behind it had me lost. Isn't the fact that the function is discontinuous mean I can't just simply use an integrating factor over the whole expression? Don't I need to first work it out like this:
when $x$ does not equal $x_0$:
$$ G(x;x_0) = \begin{cases} ae^{-x} & \text{ when }x < x_0, \\ be^{-x} & \text{ when }x > x_0. \end{cases} $$