Question

Let L, a constant, be the number of people who would like to see a newly released movie, and let N(t) be the number of people who have seen it during the first t days since its release. The rate that people first go see the movie, dN dt (in people/day), is proportional to the number of people who would like to see it but haven’t yet. Write and solve a differential equation describing dN dt where t is the number of days since the movie’s release. Your solution will involve L and a constant of proportionality k.

Answer 1

N(t) = L(1-ε^(-kt))

Explanation:

Lets call h(t) = L-N(t), the total of people that didnt get to see the movie yet. Note that h'(t) = L'-N'(t) = -N'(t) (because L is a constant).

Since N'(t) = k*h(t), we get that h'(t) = -kh(t). Therefore, we have that h(t) = c*ε^(-kt) for certain constant c. As a result, N(t) = L - h(t) = L - cε^(-kt). Its common sense that N(0) = 0, because 0 people go to see the movie before it cames out, as a consecuence we obtain that

0 = N(0) = L - c

hence, c = L, and we have then

N(t) = L - Lε^(-kt) = L(1-ε^(-kt))

I hope that works for you!