Question

A disease is spreading through a herd of 200 goats. Let G(t) be the number of goats who have the disease t days after the outbreak. The disease is spreading at a rate proportional to the number of goats who do not have the disease. Suppose that 20 goats had the disease initially and 50 goats have the disease after 2 weeks.

(a) Give the mathematical model (initial-value problem) for G.
(b) Find the general solution of the differential equation in (a).
(c) Find the particular solution that satisfies the given conditions.

Answer 1

`G(t) = 200-180e^(-0.09116t)`

Explanation:

Given that a disease is spreading through a herd of 200 goats. Let G(t) be the number of goats who have the disease t days after the outbreak. The disease is spreading at a rate proportional to the number of goats who do not have the disease. Suppose that 20 goats had the disease initially and 50 goats have the disease after 2 weeks.

a) i.e.
`G'(t) = k(200-G(t))\\dG/(200-G) = kdt\\-ln |200-G| = kt+C\\200-G = Ae^(-kt) \\G(t) = 200-Ae^(-kt)`

Initially

b)
`G(0) = 200-A =20\\A = 180\\G(t) = 200-180e^(-kt)`

c) G(2) =50

`200-180e^(-2k) =50\\15/18 = e^(-2k)\\k=0.09116`
`G(t) = 200-180e^(-0.09116t)`