Social Diffusion (Integration)

In a sufficiently large population the number x who have the information is treated as a differentiable function of time t The rate of diffusion, dx/dt, is assumed to be proportional to the number of people who have the information times the number of people who don't. This leads to the differential equation dx/dt = Kx(N-x), where N is the number of people in the population.

Suppose t is measured in days, K=1/25, and 4 people start a rumor a time t=0 in a population N=100 people.

a) Find x as a function of t by integrating both sides of the equation (1/(x(N-x))) dx = K dt
b) When will half the population have heard the rumor?
c) When will the rumor be spreading the fastest?

I'm stuck and don't know where to go.

Quote:

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Originally Posted by Preston019

I'm stuck and don't know where to go.

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Do you need help translating the word problem? Or is it the integral you don't know how to do? Or is it all just a blur?

Here's the set up, after translating the information in the word problem:

= number of people hearing the rumor at time t in days.

.

.

a) FIND . Hint: .

b) FIND such that .

c) FIND such that .