46–48. Analyzing models The following models were discussed in Section 9.1 and reappear in later sections of this chapter. In each case, carry out the indicated analysis using direction fields.
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. Let I=10mg/hr and k=0.05 hr^−1.
a. Draw the direction field, for 0≤t≤100, 0≤y≤600.

Verified step by step guidance

1. Identify the given differential equation: \(m'(t) + k m(t) = I\), where \(m(t)\) is the mass of the drug in the blood at time \(t\), \(k = 0.05\) hr\(^{-1}\), and \(I = 10\) mg/hr.
2. Rewrite the differential equation in the standard form for direction fields by isolating \(m'(t)\): \(m'(t) = I - k m(t)\) which becomes \(m'(t) = 10 - 0.05 m(t)\).
3. Understand that the direction field is a graphical representation of the slope \(m'(t)\) at various points \((t, m)\) in the plane. For each point \((t, m)\), the slope is given by \$10 - 0.05 m$.
4. Set up a grid of points for \(t\) from 0 to 100 and \(m\) from 0 to 600. At each point, calculate the slope using \(m'(t) = 10 - 0.05 m\) and draw a small line segment with that slope to represent the direction field.
5. Interpret the direction field: Notice how the slope changes with \(m\). When \(m\) is small, the slope is positive (drug mass increasing), and when \(m\) is large, the slope becomes negative (drug mass decreasing). This helps visualize the behavior of the drug concentration over time.

Key Concepts

Direction Fields

Direction fields, or slope fields, graphically represent solutions to first-order differential equations by showing the slope of the solution curve at various points. Each small line segment indicates the slope given by the differential equation at that point, helping visualize the behavior of solutions without solving the equation explicitly.

First-Order Linear Differential Equations

A first-order linear differential equation has the form y' + p(t)y = q(t). It models processes where the rate of change depends linearly on the current state and an external input. Understanding its structure is essential for interpreting the given drug infusion model and predicting how the drug mass changes over time.

Modeling Drug Infusion with Differential Equations

The drug infusion model m'(t) + km(t) = I describes how the drug mass in the bloodstream changes, balancing infusion rate and absorption. Here, k represents the absorption rate constant, and I the infusion rate. Analyzing this model helps predict drug concentration dynamics and steady-state behavior.