Question

At a time denoted as t = 0 a technological innovation is introduced into a community that has a fixed population of n people. Determine a differential equation for the number of people x(t) who have adopted the innovation at time t if it is assumed that the rate at which the innovations spread through the community is jointly proportional to the number of people who have adopted it and the number of people who have not adopted it. (Use k > 0 for the constant of proportionality and x for x(t). Assume that initially one person adopts the innovation.)
dx/dt=

x(0)=

Answer 1

N = number of people = constant

number of people who have adopted the new technology: x(t) = x

number of people who have not adopted the new technology: n - x

proportionality constant: k

dx / dt = kx(1-x)

x(0) = 0 (this is at t = 0, 0 people have adopted the new technology)