(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Deriving Logistic Population Model (ODE question)

**Physics Forums | Science Articles, Homework Help, Discussion**