Impulse Response Function-Green's Function
A simple way to derive the differential equation from the impulse response is to transform the latter to the frequency domain, rewrite the input/output relation, and then transform the resulting equation back to the time domain.

The Laplace transform of the given impulse response $h(t)$ is

$$\begin{align}
H(s) &amp;=\frac{A}{\tau}\frac{1}{\left(s+\frac{1}{\tau}\right)^2} \\
&amp;=\frac{A}{\tau}\frac{1}{s^2+\frac{2}{\tau}s+\frac{1}{\tau^2}} \\
&amp;=\frac{U(s)}{M(s)}\tag{1}
\end{align}$$

where $U(s)$ and $M(s)$ are the Laplace transforms of the output $u(t)$ and input $m(t)$, respectively.

From $(1)$ we get

$$U(s)\left[s^2+\frac{2}{\tau}s+\frac{1}{\tau^2}\right]=\frac{A}{\tau}M(s)\tag{2}$$

Transforming $(2)$ back to the time domain finally gives the desired differential equation

$$\ddot{u}(t)+\frac{2}{\tau}\dot{u}(t)+\frac{1}{\tau^2}u(t)=\frac{A}{\tau}m(t)\tag{3}$$