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.

N (t) = L (1-ε^ (-kt))

Step-by-step 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!