Question

Out of a total of N students at a school, the number of students who have seen a new television program increases at a rate proportional to the product of the number of students who have seen the program and the number of students who have not seen the program. If S denotes the number of students who have seen the program at time t, which of the following differential equations could be used to model this situation, where k is a positive constant?

a. ds/dt = KS
b. ds/dt = kt(N-t)
c. ds/st = KS(N-S)
d. ds/dt = KS(S-N)

Answer 1

ds/dt = kN(N-S)

Explanation:

Given

Total number of student = N

number of students who have seen the program = S

Number of students who have not seen the program = N-S

The product of the number of students who have seen the program and the number of students who have not seen the program = N(N-S)

If the rate of number of students who have seen a new television program increases at a rate proportional to the product of the number of students who have seen the program and the number of students who have not seen the program, then:

ds/dt = kN(N-S)

k is the proportionality constant