Differential Equations question: One person in a small community of 10,000 peopl

Question

Differential Equations question:
One person in a small community of 10,000 people gets a mild but very contagious cold. After one day four people have the cold. The cold spreads at a rate proportional to both the number of people who have had it and the number of people who have not had it. Approximately how long will it take for the entire community to have had the cold? Define this time to be the time for the number who have a cold to reach 9999.5.
Differential Equations question:
One person in a small community of 10,000 people gets a mild but very contagious cold. After one day four people have the cold. The cold spreads at a rate proportional to both the number of people who have had it and the number of people who have not had it. Approximately how long will it take for the entire community to have had the cold? Define this time to be the time for the number who have a cold to reach 9999.5.

One person in a small community of 10,000 people gets a mild but very contagious cold. After one day four people have the cold. The cold spreads at a rate proportional to both the number of people who have had it and the number of people who have not had it. Approximately how long will it take for the entire community to have had the cold? Define this time to be the time for the number who have a cold to reach 9999.5.

Explanation / Answer

let P be number of people who have cold

dP/dt = k*P(10000-P)

P(0) = 1

P(1) = 4

dP/(P*(10000-P))= kdt

(1/P + 1/(10000-P) dP = 10,000 k dt

ln P - ln (10000-P) = 10000kt + c

at t = 0 , P = 1

at t = 1 ,P = 4

-ln9999 = c

ln (4/9996) = 10000*k+c

so

when P = 9999.5 , t= ?

ln (9999.5/0.5) = 10000k*t + c

ln(19999) = (ln(4/9996) + ln (9999)) t/10000 - ln(9999)

t =137846.2037 day

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