For a system of linear equations $y'=Ay+g(t)$, my textbook gives a bound $$|\phi (t)| \leq K|y|\exp[p(t-t_0)]+K\int^{t}_{t_0} \exp[p(t-s)]\,|g(s)|\,ds.$$
I am trying to show that, under certain additional assumptions, we can improve this bound to some constant for all $t$. I am confused on how this bound can include $|y|$.
To be more specific, I have improved the bound to where it only depends on $|y|$, and nothing else. How can I further improve this bound while it relies on $|y|$, something I have never encountered?