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.
c. What is the equilibrium solution?

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 at time \(t\), \(k = 0.05\) hr\(^{-1}\), and \(I = 10\) mg/hr.
2. Recall that the equilibrium solution occurs when the mass of the drug does not change over time, meaning the derivative \(m'(t) = 0\).
3. Set \(m'(t) = 0\) in the differential equation to find the equilibrium: \$0 + k m = I$.
4. Solve for \(m\) to find the equilibrium mass: \(k m = I \implies m = \frac{I}{k}\).
5. Substitute the given values of \(I\) and \(k\) into the expression to write the equilibrium solution explicitly as \(m = \frac{10}{0.05}\) mg.

Key Concepts

Differential Equations and Their Solutions

A differential equation relates a function with its derivatives, describing how the function changes over time. Solving it involves finding a function that satisfies this relationship, often representing real-world processes like drug concentration in the bloodstream.

Equilibrium Solution of a Differential Equation

An equilibrium solution is a constant solution where the derivative is zero, meaning the system is in a steady state. For the equation m'(t) + km(t) = I, the equilibrium occurs when the rate of change m'(t) = 0, balancing drug infusion and absorption.

Direction Fields (Slope Fields)

Direction fields graphically represent the slopes of solutions to a differential equation at various points. They help visualize the behavior of solutions over time and identify equilibrium solutions by showing where slopes are zero or constant.