Population increases are discouraged as population tends towards M.

dP/dt = kP(M - P)

integral( 1 / (P*(M - P), dx) = integral(k, dt)

...

P / (M - P) = e^(Mkt) + C

(1)
Is this correct?

(2)
How do you reduce this into a function that shows P and P(0) instead like it is shown in the book? I have scowered the internet and all I've seen is that it takes algebra.

Basically. It should be dP in the first integral. But otherwise this seems fine.

Edit: Ooops... Misread your parentheses... Your constant of integration should be up in your exponent.

There are a couple ways.

a) When you integrate, use definite integral on both sides. Your dt integral should go from 0 to t and your dP integral should go from P_0 to P. (Technically, you should change the variables so that you're not integrating with respect to the variable in the limits... but that's one of those things that you can usually just get away with.)

b) When you integrate, use the indefinite integral, putting your constant of integration on the right side. Right after you integrate, set t = 0 and P = P_0. This should give you an equation that you can solve for C in terms of P_0. Then plug that in and continue solving for P. (If you wait until the end to make your substitution, the algebra gets worse.)

Edit: Actually, since you're just exponentiating, it doesn't get that much worse. You just simplify the right side like this: e^{Mkt + C} = e^C * e^{Mkt} = C' e^{Mkt}.

https://en.wikipedia.org/wiki/Logistic_function

To do the integral

Int( dp/(p*(M - p))) you can write 1/(p*(M-p)) as 1/M*(1/p+1/(M-p)) and then its simple logs.