Chapter 9.1, Problem 52E

Drug infusion The delivery of a drug (such as an antibiotic) through an intravenous line may be modeled by the differential equation m′(t) + km(t) = I, where m(t) is the mass of the drug in the blood at time t ≥ 0, k is a constant that describes the rate at which the drug is absorbed, and I is the infusion rate.

a.    Show by substitution that if the initial mass of drug in the blood is zero (m(0) = 0), then the solution of the initial value problem is $m\left(t\right)=\frac{I}{k}\left(1-e^{-kt}\right)$.

b.    Graph the solution for I = 10 mg/hr and k = 0.05 hr^–1.

c.    Evaluate $\underset{t\to \infty }{\lim }m\left(t\right)$, the steady-state drug level, and verify the result using the graph in part (b).