# Differential equations - logistic law problem

1. A population grows accordin to the logistic law, with a limiting population of 5 x 10^8 individuals. When the population is low, it doubles every 40 minutes. What will the population be after 2 hours if initially it is (a) 10^8, and (b) 10^9?

P=P0e^a(t-t0)
N=a/b
p(t) = ap0/[bp0 + (a-bp0)e^-a(t-t0)]

I have actually solved the problem, but I want to confirm that I am doing it right so here is my solution, i only showed (a) since a and b are the same.

p=p0e^a(t-t0) (i used this since it gives you info about a low population)
2=1e^a(40-0)
2=e^40a
40a = ln 2
a = 1/40 ln 2

then i used N = a/b since N is given to be 5 x 10^8
5 x 10^8 = a/b
5 x 10^8 = (1/40 ln 2) / b
b = (1/40 ln 2)/(5 x 10^8)

then i used the logistic model
p(t) = ap0/[bp0 + (a-bp0)e^-a(t-t0)]
p(120)= (1/40 ln 2)(10^8)/[((1/40 ln 2)/(5x10^8)) (10^8) + (1/40 ln 2 - ((1/40 ln 2)/(5x10^8)) (10^8) )e^(-1/40 ln 2)(120-0)] (sorry i know this looks confusing bare with me)
P(120) = (10^8/40 ln 2) / [1/8 ln 2 + (1/40 ln 2- (1/8 ln 2))e^-3 ln 2)]
P(120) = (10^8/40 ln 2) / [1/8 ln 2 + (1/40 ln 2 - (1/8 ln 2))(1/8)] (e^-3 ln 2 = 1/8, is this correct??)
P(120) = (10^8/40 ln 2) / [1/8 ln 2 + 1/320 ln 2 - 1/64 ln 2]
then i got rid of all the ln 2's
P(120) = (10^8/40) / [1/8 + 1/320 - 1/64]
P(120) = (10^8/40) / 9/80
P(120) = (10^8/40)(80/9)
P(120) = (2 x 10^8)/9 = 22,222,222

i really hope this is right.
and i did the same thing for (b) but changed 10^8 to 10^9 and got 22,792,022