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## Homework Statement

Find the Green's function $G(t,\tau)$ that satisfies

$$\frac{\text{d}^2G(t,\tau)}{\text{d}t^2}+\alpha\frac{\text{d}G(t,\tau)}{\text{d}t}=\delta(t-\tau)$$

under the boundary conditions $$G(0,\tau)=0~~~\text{ and }~~~\frac{\text{d}G(t,\tau)}{\text{d}t}=0\big|_{t=0}$$

Then, solve $$\frac{\text{d}^2x(t)}{\text{d}t^2}+\alpha\frac{\text{d}x(t)}{\text{d}t}=f(t)$$ for $$f(t)=\begin{cases}0&\text{if }~t<0\\Ae^{-\beta t}&\text{if }~t\geq 0\end{cases}$$

## The Attempt at a Solution

By solving the homogenous equation and letting it vary for ##t<\tau## and ##t>\tau## due to the discontinuity at ##t=\tau## in the first derivative of ##G(t,\tau)## I get:

$$G(t,\tau)=\begin{cases}A(\tau)e^{-\alpha t}+B(\tau)&\text{if }~t<\tau\\C(\tau)e^{-\alpha t}+D(\tau)&\text{if }~t> \tau\end{cases}$$

The way my teacher did it, he applied the boundary equations to the case where ##t<\tau## and deduced that ##A(\tau)=B(\tau)=0##. When I try solve this question, it only makes sense if I apply the boundary conditions to the case where ##t>\tau## since ##f(t)## is defined for ##-\infty<t<\infty## and the boundary conditions are given at ##t=0##. What I am trying to say, is I can never tell whether I am meant to apply the boundary conditions to the case where ##t<\tau## or ##t>\tau##, I always feel like I am guessing which one to choose. I want to develop some intuition for this part, the rest of the question I can do. I think this might have to do with the fact that we can choose the value for which ##\tau## satisfies the boundary conditions, but I am still not sure. Cheers.