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## Homework Statement

Solve the logistic population model:

[itex] dP/dt=rP(1-P/C); P(0)=P_{0}[/itex]

**2. The attempt at a solution**

First, I separated variables to get:

[itex]\int \! \frac{1}{P(1-P/C)} \, \mathrm{d}P = \int \! r \, \mathrm{d}t[/itex]

Then, I took the left hand side and split into partial fractions:

(1) - [itex]\int \! \frac{1}{P} \, \mathrm{d}P + \int \! \frac{1/C}{1-P/C} \, \mathrm{d}P[/itex]

If I integrate, I get the following:

[itex]\ln(P)-\ln(1-P/C)=\ln(\frac{P}{1-P/C})[/itex] (*)

However, my problem is this. If I take (1) and multiply the second integral by C/C (which should be fine, its 1), I get the following:

[itex]\int \! \frac{1}{P} \, \mathrm{d}P + \int \! \frac{1}{C-P} \, \mathrm{d}P[/itex]

Which is...

[itex]\ln(P)-\ln(C-P)=\ln(P/C-P)[/itex] (**)

However, (*) and (**) are not the same. I'm assuming there's something wrong with one of the ways that I integrated considering I came out with two different functions. Are they both correct and all that will change when I do the full problem out is the constant?