I'm having trouble with the following textbook problem:

The only basic approach for solving differential equations that I know is to "separate and integrate", that is, have one side with P's and a dP, and the other side with t's and a dt (constants anywhere throughout), and then integrate both sides. So, what I was hoping to get was kM * something * dt = kP * something * dP. However, it looks like that's not going to happen. Could someone help me solve this thing?

I know that if "M is the maximum level of performance", then logically [tex]\lim_{t \rightarrow \infty} P(t) = M[/tex]. However, I'm still not sure how to actually find the function P(t).

Ah, of course! I guess my first mistake was distributing the k. Dividing brings me to [tex]\frac{dP}{M - P} = kdt[/tex], and integrating both sides gives me [tex]-ln|M - P| = kt + c[/tex]. Going from there:

[tex]ln|M - P| = -kt - c[/tex]
[tex]e^{-kt - c} = M - P[/tex]
[tex]P = M - e^{-kt - c}[/tex]
I think I'm doing something very wrong here. The answer in the back of the book is [tex]P(t) = M - Me^{-kt}[/tex].

I'm thinking I've made a really stupid mistake, but I can't find it. Can anyone help?