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

I know the answer for the first one: dx/dt = kx(n-x) but I'm not sure how to do the 2nd one.

Answer 2

you mean ... x(0) ?

they give you initial value of 1
x(0) = 1

Answer 3

As far as solving the diff equation, separate variables
\[\rightarrow \frac{dx}{x(N-x)} = k dt\]
integrate left side using partial fractions:
\[\frac{A}{x} + \frac{B}{N-x} = \frac{1}{x(N-x)}\]
\[A = B = \frac{1}{N}\]
\[\frac{1}{N} \int\limits \frac{dx}{x} + \frac{1}{N} \int\limits \frac{dx}{N-x} = k \int\limits dt\]
\[\frac{1}{N} (\ln x - \ln (N-x)) = kt + C\]
\[\ln \frac{x}{N-x} = Nkt + C\]
\[\frac{x}{N-x} = e^{Nkt +C}\]
\[x = Ne^{Nkt +C} - x e^{Nkt+C}\]
\[x = \frac{N e^{Nkt +C}}{1 + e^{Nkt +C}}\]

Answer 4

How did you get x(0) = 1?