A drug is administered intravenously to a patient at a rate of 12 mg per day. About 40% of the drug in the patient's body is metabolized and leaves the body each day. Which is the differential equation for the amount of the drug, D, in the body as a function of time, t, in days?

Question

Quizplus · Accepted Answer

Let D(t) be the amount of drug in the body at time t. Then the rate of change of D with respect to time is the difference between the amount of drug entering the body and the amount leaving due to metabolism. 
The amount of drug entering the body is constant at 12 mg per day. 
The amount leaving the body due to metabolism is 40% of the drug in the body, which is 0.4D(t). 
Therefore, the differential equation is:
dD/dt = 12 - 0.4D(t)
which can be rearranged as:
dD/dt + 0.4D(t) = 12
This is a first-order linear differential equation, which has the general solution:
D(t) = Ce^(-0.4t) + 30
where C is the constant of integration. 
Since the initial condition is not given, we cannot determine the value of C.

A Drug Is Administered Intravenously to a Patient at a Rate