Question

Infusion of a Drug A drug is infused into a patient’s bloodstream at a constant rate of r grams per second. Simultaneously, the drug is removed at a rate proportional to the amount x(t) of the drug present at time t. Determine a differential equation for the amount x(t).

Short answer

The differential equation is $\frac{dx}{dt}=r-c·x\left(t\right)$.

Step-by-step solution

Definition of Newton’s law of cooling theith mathematical statement

Newton’s law of cooling/warming translates into the mathematical statement

$\frac{dT}{dt}\propto T-T_m\text{or}\frac{dT}{dt}=k\left(T-T_m\right)$

Where k is a constant of proportionality. In either case, cooling or warming, if $T_m$is a constant, it stands to reason that $k<0$.

Compute Differentiation

Now, we can use the fact that the rate of drugs in blood is determined by the expression $\frac{dx}{dt}=r-R$.

Where r is rate of drug increase, and rate of drug decrease.

Since rate at which drug is removed is proportional to amount of the drug at present time, we can write

$R=c·x\left(t\right)$

Where is constant, $c\ne 0$.

Therefore, the Differential equation of the amount x(t) will be

$\frac{dx}{dt}=r-c·x\left(t\right)$.