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!
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 $p(5) = N/2$.