There's a passage on Ogata's Modern Control Theory on dealing with the response of a system to initial conditions where he does the following and then says this about it:
I can't see any sense in this passage. We assume at first arbitrary initial conditions (which is what we're interested in), which yield
$$\mathscr L[\dot{\mathbf x}](s)=s\mathbf{X}(s)-\mathbf x(0)$$
from which it is immediate that
$$s\mathbf{X}(s)=\mathscr L[\dot{\mathbf x}](s)+\mathbf x(0)$$
so that, taking inverse Laplace Transforms for both sides
$$\mathscr L^{-1}[s\mathbf{X}(s)]=\dot{\mathbf x}+\mathbf x(0)\delta$$
At least this is what I assume to be true.
In equation 5-50, however, when the author takes the inverse Laplace transforms, he neglects the delta term, as if now the initial conditions are all zero, so that
$$\mathscr L^{-1}[s\mathbf{X}(s)]=\dot{\mathbf x}$$
and arrives at equation 5-51, finally asserting that "taking the Laplace transform of a differential equation and then by taking the inverse Laplace transform of the Laplace-transformed equation we generate a differential equation that involves the initial condition".
I can't see how any of this makes any sense. What am I missing? If we take the Laplace transform of the functions on both sides of an equation and then take the inverse Laplace transforms of the Laplace-transformed equation, I could only imagine we would obtain the original equation. Instead, in his exposition, it seems like at first the arbitrary initial conditions are taken into account, and then suddenly they are not. What gives?