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.

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.[tex]G'(t) = k(200-G(t))\\dG/(200-G) = kdt\\-ln |200-G| = kt+C\\200-G = Ae^{-kt} \\G(t) = 200-Ae^{-kt}[/tex]

Initially

b) [tex]G(0) = 200-A =20\\A = 180\\G(t) = 200-180e^{-kt}[/tex]

c) G(2) =50

[tex]200-180e^{-2k} =50\\15/18 = e^{-2k}\\k=0.09116[/tex][tex]G(t) = 200-180e^{-0.09116t}[/tex]