Question

At a time denoted as t = 0 a technological innovation is introduced into a community that has a fixed population of n people.

1. 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.)

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

The differential equation is

dR/dt = -k(2x - n)dx/dt for k > 0

Assuming initially, one person adopts the innovation, then

dR/dt = 0

Explanation:

Total number of people in the community is "n"

At time t, the number of people who

have adopted the innovation is "x(t)"

This Tells us that (n - x) people haven't adopted the innovation.

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.

So

Let R be the rate, then

R is proportional to x(n - x)

R = kx(n - x) = -kx(x - n)

Differentiating this with respect to time, t, we have

dR/dt = (-k(x - n) - kx)dx/dt

dR/dt = -k(2x - n)dx/dt for k > 0

And this is the differential equation.

Assuming initially, one person adopts the innovation, then

dR/dt = 0