Impulse Response and Linearity Lately in my physics class (2nd-year undergraduate) we've been learning about the impulse response of systems and using Green's Functions to model the response of a system to more complicated input forces. Since it's a physics class, not a math class, some of the arguments for the math have been a bit handwavy. I've been going back through the work, then, to try to convince myself of all of the steps. I'm on board with everything up until the final point:
To get the response of a system to a complicated input force, you can find the response of the system to a series of impulses, then add together the resulting responses to get the total response. This requires the assumption that the motion of a system as a function of the given force is linear, i.e.:
$$y(F_1(t))+y(F_2(t))=y((F_1+F_2)(t))$$
Where $y$ is the system's position at some given time, and $F_1(t)$ and $F_2(t)$ are different time-dependent forces applied to the system. More visually, you need the assumption that, in the following picture, it's legitimate to get the actual motion of the system by adding together the two dashed red lines, each of which is the response to an impulse at a different time.

Now, this is obviously correct. The trouble is, I can't see why, mathematically speaking, it should be. My question, then, is how we actually prove that this is the case. I'm sure it's a simple observation that I'm just missing, but I can't for the life of me figure out right now.
 A: Maybe you can think about it like this: the equation of motion for a physical system (here, an object) which obeys Newtonian mechanics is
$$F(t) = m\frac{\mathrm{d}^2y}{\mathrm{d}t^2}$$
That is, given $F(t)$ and a fixed set of initial conditions, you can determine $y(t)$.
Now suppose you have two arbitrary functions $F_1(t)$ and $F_2(t)$. For each of these functions, imagine subjecting the object to a force that follows that function, assuming the same initial conditions, and then using the above equation to determine the resulting motion of the object, $y_1(t)$ and $y_2(t)$ respectively.
$$\begin{align}
F_1(t) &= m\frac{\mathrm{d}^2 y_1}{\mathrm{d}t^2} \\
F_2(t) &= m\frac{\mathrm{d}^2 y_2}{\mathrm{d}t^2}
\end{align}$$
By adding these two equations, you get
$$F_1(t) + F_2(t) = m\frac{\mathrm{d}^2(y_1 + y_2)}{\mathrm{d}t^2}$$
which shows that $y_1(t) + y_2(t)$ is the motion that results from the combined time-dependent force $F_1(t) + F_2(t)$.
If you're analyzing some physical system that is more complicated than a simple object, then the equation of motion may be something more complicated than $F_\text{net} = ma$. But as long as the underlying equation of motion specifies a linear relationship between $y(t)$ and $F(t)$, then you can use Green's functions.
