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while studying for my math exam coming up, I stumbled upon this exercise.

A certain product is sold on a market with 1 000 000 potential customers. We assume that the number of people p that have bought the product is growing logistically. So, the function p(t) (where time t is measured in years) satisfies a differential equation of the form $$\frac{dp}{dt}= \frac{k}{N} p(N - p)$$$, with N the value of p in the long run and k some positive number. It is expected that in three years’ time, a quarter of the potential customers will have acquired the product, that in five years’ time, half of them will and in the long run everybody will.

If I understand correctly we can say that p(3) = 1/4 * 1 000 000 ,
p(5) = 1/2 * 1 000 000 , N = 1 000 000

The question asked is : Find the equation of the function p(t) I can't seem to find the right solution Can somebody enlighten me? Many thanks!

Queen Wayfarer
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  • Welcome to MSE. It is in your best interest that you use MathJax. – José Carlos Santos Jun 10 '18 at 09:51
  • "putting in these numbers in the equation" has little meaning. What you need to do is to solve the differential equation, and then to use the values that are given to you to identify its parameters. (Note additionally that, at present, there is no question in your post.) – Did Jun 10 '18 at 10:14
  • How to solve the ODE: https://math.stackexchange.com/questions/78560/how-do-you-solve-the-initial-value-probelm-dp-dt-10p1-p-p0-0-1 – Hans Lundmark Jun 10 '18 at 13:48

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To solve the differential equation, rewrite it as $$kdt = {dp\over p} + {dp\over{N - p}}.$$ By integrating and re-arranging we get $p = N/(1 + ce^{-kt})$, where $c$ is a constant. The values of $c$ and $k$ can then be found from the information that $p(3) = N/4$ and $p(5) = N/2$.

Michael Behrend
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