Suppose a student carrying a flu virus returns to an isolated college campus of 5000 students. Determine a differential equation governing the number of students x (t) who have contracted the flu if the rate at which the disease spreads is proportional to the number of interactions between students with the flu and students who have not yet contracted it. (Use k > 0 for the constant of proportionality and x for x (t).)

x (t) = 5000 * (1 - e^-kt)

Step-by-step explanation:

Given:

- Total number of students n = 5000

Find:

Differential equation governing the number of students x (t) who have contracted the flu.

Solution:

- Number of non-affected students = (5000 - x)

Hence,

- Rate at which students are infected:

dx / dt = k * (5000 - x)

- separate variables:

dx / (5000 - x) = k*dt

- Integrate both sides:

- Ln (5000 - x) = kt + C

- Evaluate C for x = 0 @ t = 0

- Ln (5000 - 0) = k*0 + C

C = - Ln (5000)

- The solution to ODE is:

Ln (5000 - x) = - k*t + Ln (5000)

5000 - x = 5000*e^-kt

x (t) = 5000 * (1 - e^-kt)