calculus – Assuming limits in a linear derivative equation

enter image description here

Hi, I don’t understand the reasoning behind the limits of $0<= P(0) <= M$, and for the k>0, I only managed to figure out that K can’t be equaled to 0. But maybe it’s because I did it in a different method from the book? I used the same $I(t)$ so got up to:

$e^{kt}p = KMe^{Kt}$

$e^{kt}p = int Ae^{Kt}$, where $A = KM$

$e^{kt}p = frac AK e^{Kt}$, doing a u sub where $u=Kt$

From there, if you sub in KM into A, K can’t be 0. But how did they assume the rest? Thanks