Question

At a time denoted ast = 0a 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. (Usek > 0for the constant of proportionality and x forx(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

Answer and Explanation:

He rate at which the innovation spread through the community is the number of people proportional to the people adopted it and the people who don't adopted it as constant of proportionality is k , and the fixed population of the community is n.

Suppose

Number of people who have adopted innovation at time t = x (t)

Number of people who don’t adopted innovation at time t = n – x (t)

Then the differential equation is:

dx/dt = kx (n-x), k > 0

Where k is the constant of proportionality.