At a time denoted as t = 0 a technological innovation is introduced into a commu

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 0 for the constant of proportionality and x for x(t). Assume that initially one person adopts the innovation.) dx dt x(0) =

Explanation / Answer

Solution :- Let N is the (constant) total number of people in the population, and x(t) is the number of people at time t that have adopted the innovation, then (N - x(t)) is the number of people at time t who have not adopted it.

Further The rate of change in x(t) is jointly proportional to x(t) and (N - x(t)), which means it imust be proportional to their product. therefore In mathematical terms, we can write it as

dx(t)/dt = k*x(t)*(N - x(t)) , k >0

where k is the constant of proportionality.

With initial condition , we assume that initially one person adopts the innovation ,

x(0) = 1

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