Hi all, I have a first order separable differential equation that I find a little difficult to solve. The question is from an old maths exam paper from my country. The book I obtained it from only has the answer, but I'd really like to know how to obtain the correct general solution. Okay, here is the question :(adsbygoogle = window.adsbygoogle || []).push({});

1. A large population of N individuals contains, at time t = 0, just one individual with a contagious disease. Assume that the spread of the disease is governed by the equation

[tex]\frac{dN}{dt} = k (N-n)n[/tex]

where [tex] n(t) [/tex] is the the number of infected individuals after a time t days and k is a constant.

a) Find n explicitly as a function of t.

Part a) gives me a hint telling me to rewrite [tex]\frac{1}{[(N-n)n]}[/tex] as the partial fractions [tex]\frac{1}{N}(\frac{1}{N-n} + \frac{1}{n})[/tex]. I manage to work this out, and I obtain this :

[tex]\int \frac{1}{N}(\frac{1}{N-n} + \frac{1}{n})dN = \int kdt[/tex]

But I got stuck here as I'm not sure how to go about this as all attempts give me more than one form of n, one of n and one of e to the power of n. I had a look at the back of the book and the answer gives :

[tex]n = \frac{Ne^{Nkt}}{N - 1 + e^{Nkt}}[/tex]

So this gave me some help, so in the end I had this :

[tex]\int (\frac{1}{N-n} + \frac{1}{n})dN = \int Nkdt[/tex]

But I am not sure how to get just one n term because I don't get anywhere near the result that I am looking for when I integrate both sides.

I appreciate any helpful advice on this. Thanks in advance for any correspondence.

**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: Differential Equation Separable Variables

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