A certain drug is being administered intravenously to a hosp

A certain drug is being administered intravenously to a hospital patient. Fluid containing k = 3 mg/cm^3 of the drug enters the patient\'s bloodstream at a rate of r = 70 cm^3/hr. The drug is absorbed by body tissues or otherwise leaves the bloodstream at a rate proportional to the amount present, with a rate constant of p = 0.14 (hr)^-1. Assuming that the drug is always uniformly distributed throughout the bloodstream, write a differential equation for the amount of the drug that is present in the bloodstream at any time. Your answer should be an expression in terms of q (in milligrams) and t (in hours). dq/dt = How much of the drug is present in the bloodstream after a long time?

Solution

Let Q(t) represent the quantity of the drug in the bloodstream, measured in mg, and let t represent time, measured in hours.

Then

dQ/dt = rate in - rate out

dQ/dt = 3(70) - (0.14)Q

dQ/dt = 210 - (0.14)Q

dQ/dt + (0.14)Q = 210

we have p(t) = 0.14 and r(t) = 210

I.F = e ^{p(t) dt} = e ^{0.14dt =} e^0.14t

The general solution is

Q(I.F) = (I.F) . r(t) dt + c

Q(e^0.14t) = (e^0.14t) .210 dt + c

Q(e^0.14t) =210 . (e^0.14t) / ( 0.14) + c

Q(e^0.14t) = 1500(e^0.14t) + c

Q = 1500 + c (e^-0.14t)

Since Q(0) = 0

Q(0) = 1500 + c (e^-0.14(0))

0 = 1500 + c thus c = - 1500

Therefore , the quantity of drug in the bloodstream at any time is

Q(t) = 1500 - 1500 (e^-0.14t)

After a long time that is for t is tends infinity Q( infinity) = 1500 mg