Error in Gilbert Strang Differential Equations 2.1 Fundamental Solution to 2nd Order Differential Equation? In Strang's Differential Equations and Linear Algebra book, Section 2.1, he introduces the fundamental solution / impulse response $g(t)$ to the delta forcing function:
$$m g'' + kg = \delta(t) $$
with given initial conditions $g(0) = 0$ and $g'(0) = 0$. He then says this implies that $g'(0) = 1/m$, since
$$ m g''(0) + k g(0) = \delta(t) \rightarrow m g''(0) = \delta(t) \rightarrow m g'(0) = 1 \rightarrow g'(0) = \frac{1}{m}$$
I'm confused how this doesn't contradict the initial given condition $g'(0) = 0$. Could someone please clarify?
Edit 1: Maybe I'm misunderstanding the text. I don't find Strang easy to follow. Photo below:

 A: I agree that Strang didn't do a great job here... I'm particularly disappointed by his lack of clarification on the domains of his solutions, since this one in particular is implied to be the solution for positive time only. Anyways, on to answering what is happening here (and sorry I'm so late to the game):
For functions with the step function $H(t)$ or the impact function $δ(t)$, there is a strange change in $y$ at $t = 0$ (for $H(t)$, there's a jump from $0$ to $1$, and $δ(t)$ has an instantaneous impact), but besides that point the equation is pretty run-of-the-mill.
For a differential equation $Ay'' + By' + Cy = δ(t)$, it is not inaccurate to say that for $t ≠ 0$, the equation is just the homogeneous version of itself: $Ay'' + By' + Cy = 0$. However, the instantaneous impact at $t = 0$ of course has some effect on the equation, and it can't just be said that we can replace it with zero. But we are certain with this approach that the solution $y$ of $Ay'' + By' + Cy = δ(t)$ for $t < 0$ is a solution of the homogeneous equation above, and that the solution for $t > 0$ is another solution of the homogeneous equation, but is different than that for the solution of $t < 0$ because of the impact. In other words, the effect of the delta function here is to change the initial conditions of the solution.
So, Strang is saying that given initial rest conditions $g(0)=g'(0)=0$ for $mg'' + kg = δ(t)$, the solution specifically for the domain $t > 0$ is identical to the solution of $mg'' + kg = 0$ with $g(0)=0$ and $g'(0)=\frac{1}{m}$.
Since I'm so late to answering this, I can only hope to help someone else who stumbled across this.
