公衛系 微積分期末考
(28/12/2018)
1. Use the Laplace transform to solve the differential equations.
(1) j(t)+2y(t) = x(t), y(0)=1, x(t)=10, t20 (20)
(2) Intravenous glucose is a treatment. Disposed at a fixed rate k grams per minute inputs into the blood,
while blood glucose will be converted to other substances or moved to another place, at a rate
proportional to the amount of glucose in the blood, the proportionality constant is a (a> 0), the initial
amount of glucose in the blood is M.
A. Find the variation in the amount of glucose in the blood (15)
B. Determining the equilibrium, the amount of glucose in the blood. (5)
=
2. SI Epidemic Model : The size of the population, n+1, remains fixed. Let i(t) be the number of infectives
at time t, and let s(t) be the number of individuals who are susceptible. Given an initial number of
infectives iO), we would like to know what will happen to i(t). SI Epidemic Model is described by the
differential equation.
di(t)
= k·i(t).s(t) ......(5.1)
dt
i(t)+s(t)=n+1
i(0)=i,
(1) Solve this differential equation of the SI Epidemic Model (5.1). (10 h)
(2) What is the peak times t of the epidemic spread? (10)
3. Consider the Two-compartment physiological models and is shown in figure 1. C1 (t) represent the
drug concentration in the first compartment and C2 (t) represents the drug concentration in the second
compartment. Vi and V2 represent the compartment volume. Use the first order linear differential
equation general solution to solve the C1 (t) (20 ) and use the Laplace transform to solve C2 (t).
【20 分).
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