52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.


{Use of Tech} 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.


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

Verified step by step guidance

1. Identify the given differential equation: \(m'(t) + k m(t) = I\), where \(I = 10\) mg/hr and \(k = 0.05\) hr\(^{-1}\).
2. Recognize that this is a first-order linear ordinary differential equation. The standard approach is to find an integrating factor to solve it.
3. Calculate the integrating factor \(\mu(t) = e^{\int k \, dt} = e^{k t}\), which in this case is \(e^{0.05 t}\).
4. Multiply both sides of the differential equation by the integrating factor to get \(e^{0.05 t} m'(t) + 0.05 e^{0.05 t} m(t) = 10 e^{0.05 t}\), which simplifies to \(\frac{d}{dt} \left( e^{0.05 t} m(t) \right) = 10 e^{0.05 t}\).
5. Integrate both sides with respect to \(t\) to find \(e^{0.05 t} m(t) = \int 10 e^{0.05 t} dt + C\), then solve for \(m(t)\) by dividing both sides by \(e^{0.05 t}\). This will give the general solution for \(m(t)\), which you can then graph.

Key Concepts

First-Order Linear Differential Equations

These are differential equations of the form m'(t) + p(t)m(t) = q(t), where the derivative of the function and the function itself appear linearly. The given drug infusion model fits this form, allowing the use of standard methods like integrating factors to find explicit solutions.

Integrating Factor Method

This technique solves first-order linear differential equations by multiplying both sides by an integrating factor, typically e^(∫p(t)dt), which simplifies the equation into an exact derivative. Applying this method to m'(t) + km(t) = I helps find the general solution for the drug mass over time.

Interpretation and Graphing of Solutions

Once the solution m(t) is found, understanding its behavior over time is crucial. For the drug infusion model, the solution typically approaches a steady-state value, reflecting equilibrium between infusion and absorption. Graphing with given parameters (I = 10, k = 0.05) visualizes this dynamic.