I am studying Feedback Control of Computing Systems. (specifically using Hellerstein's book, section 3.1.4, page 76)
To solve difference equation Z-Tranform can be applied. In the book there is an example I did not understand how the solution was deduced. Given an integrator: (linear difference equation)
$$ y(k+1) = y(k) + u(k) $$
and the corresponding Z-transform:
$$\sum_{k=0}^{\infty}{y(k+1)z^{-k}} = zY(z) - zy(0)$$
So (begin)
$$ zY(z)-zy(0)=Y(z)+U(z)$$
Solving it: (end)
$$ Y(z) = \frac{1}{z-1}[U(z)+zy(0)] $$
What are the intermediate steps from the equation labeled as begin through the end one?
Assumption: all signals have a value of $0$ for $k<0$.