Q:

General Solution to External Force with Damping (Synergetics by Haken Self-Organization)

In Chapter 7 Section 1 of Hermann Haken's textbook Synergetic we are given a differential equation of the form:
$$ \dot{q}=-\gamma q + F(t)$$
where $\gamma > 0$ can be interpreted as a damping constant and $F(t)$ can be interpreted as an external force applied to the system.  It is then claimed without justification that the solution to this differential equations is
$$q(t)=\int_0^t e^{-\gamma(t-\tau)}F(\tau)d\tau$$
In order to show this is true I considered just taking the derivative of $q(t)$ with respect to time and verifying that it indeed satisfies the differential equation.  A naive application of The Fundamental Theorem of Calculus is apparently inappropriate here (I believe because the integrand depends on both $t$ and $\tau$?). 
What is a proper way to show that this indeed is a solution to the differential equation?  My postdoc said this has something to do with exponential smoothing and moving averages, so bonus points if you can clarify that connection. :) 
Thanks! 

A:

$$\dot q+\gamma q=F(t)\implies\\
e^{\gamma t}\dot q+\gamma e^{\gamma t} q=F(t)e^{\gamma t}\implies\\
(e^{\gamma t}q)^\cdot=F(t)e^{\gamma t}\implies\\
e^{\gamma t}q(t)-q(0)=\int_0^t F(\tau)e^{\gamma \tau}d\tau\implies\\
q(t)=q(0)e^{-\gamma t}+e^{-\gamma t}\int_0^tF(\tau)e^{\gamma \tau}d\tau. 
$$
(To see how you can directly check you solution see Leibniz integral rule)

