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.