2. A patient is taking a medicine that has a half-life of 6 hours in the human body, meaning that the rate medicine is broken down is proportional to the amount present with constant of proportionality In 2 6 k = Starting at t= 0, they are on an IV drip that delivers a constant r mg per hour of medicine. Therefore, the differential equation satisfied by Q(t), the amount (in mg) of medicine in the patient's bloodstream at time t (in hours) is and the initial value problem is dQ dt dQ dt = 7- In 2 =7- Q(0) = 0. Q₂ In 2 6 -Q₂ (a) Solve the initial value problem. (b) Suppose that the medicine requires 10 mg to be in the bloodstream to be effective. What values of r are appropriate if we want the patient to have at least 10 mg in their blood within three hours? (c) Unfortunately the medicine is toxic if there are ever 100 mg in the body. Keeping in mind we still want the patient to have effective levels after three hours, which values of r are also safe if the patient remains on the IV indefinitely?
2. A patient is taking a medicine that has a half-life of 6 hours in the human body, meaning that the rate medicine is broken down is proportional to the amount present with constant of proportionality In 2 6 k = Starting at t= 0, they are on an IV drip that delivers a constant r mg per hour of medicine. Therefore, the differential equation satisfied by Q(t), the amount (in mg) of medicine in the patient's bloodstream at time t (in hours) is and the initial value problem is dQ dt dQ dt = 7- In 2 =7- Q(0) = 0. Q₂ In 2 6 -Q₂ (a) Solve the initial value problem. (b) Suppose that the medicine requires 10 mg to be in the bloodstream to be effective. What values of r are appropriate if we want the patient to have at least 10 mg in their blood within three hours? (c) Unfortunately the medicine is toxic if there are ever 100 mg in the body. Keeping in mind we still want the patient to have effective levels after three hours, which values of r are also safe if the patient remains on the IV indefinitely?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:2. A patient is taking a medicine that has a half-life of 6 hours in the human body, meaning that the
rate medicine is broken down is proportional to the amount present with constant of proportionality
In 2
6
Starting at t = 0, they are on an IV drip that delivers a constant r mg per hour of medicine.
Therefore, the differential equation satisfied by Q(t), the amount (in mg) of medicine in the patient's
bloodstream at time t (in hours) is
k
=
and the initial value problem is
dQ
dt
=7-
dQ
dt
Q(0) = 0.
In 2
=7-
Q₂
In 2
6
-Q₂
(a) Solve the initial value problem.
(b) Suppose that the medicine requires 10 mg to be in the bloodstream to be effective. What
values of r are appropriate if we want the patient to have at least 10 mg in their blood within
three hours?
(c) Unfortunately the medicine is toxic if there are ever 100 mg in the body. Keeping in mind we
still want the patient to have effective levels after three hours, which values of r are also safe
if the patient remains on the IV indefinitely?
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