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

dx/dt = kx(n-x)

x(0) = 1

Reasoning:

k is the proportional constant, so it's always there.

x represents the amount of people who have the technology.

n represents the total amount of people

to get the number of people who haven't adopted it yet, we have to take the total population of the community and subtract it from the people who do have the technology.

you can think of it like:

not adopted + adopted = total population

so, to represent everyone,

k * (number of people w/ technology) * (total population - adopted)

x(0) equals 1 because they say "Assume that initially one person adopts the innovation. At time = 0, one person has it.

Answer 2

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)