Ch. 9 - Differential Equations

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 9 - Differential Equations

Problem 9.2.46c

Chapter 9, Problem 9.2.46c

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?

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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.

Recall that the equilibrium solution occurs when the mass of the drug does not change over time, meaning the derivative \(m'(t) = 0\).

Set \(m'(t) = 0\) in the differential equation to find the equilibrium: \(0 + k m = I\).

Solve for \(m\) to find the equilibrium mass: \(k m = I \implies m = \frac{I}{k}\).

Substitute the given values of \(I\) and \(k\) into the expression to write the equilibrium solution explicitly as \(m = \frac{10}{0.05}\) mg.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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Understanding Slope Fields

Related Practice

Textbook Question

33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.

c. Repeating parts (a) and (b) using half the time step used in those calculations, again find an approximation to y(T).

y′(t) = 6 - 2y, y(0) = -1; Δt = 0.2, T = 3; y(t) = 3 - 4e⁻²ᵗ

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Textbook Question

{Use of Tech} Logistic equation for an epidemic When an infected person is introduced into a closed and otherwise healthy community, the number of people who contract the disease (in the absence of any intervention) may be modeled by the logistic equation

dP/dt=kP(1−P/A),P0=P_0,

where K is a positive infection rate, A is the number of people in the community, and P0 is the number of infected people at t=0. The model also assumes no recovery.

c. For a fixed value of K and A, describe the long-term behavior of the solutions, for any P0 with 0<P0<A.

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Textbook Question

17–20. Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed.

c. Which initial conditions y(0) = A lead to solutions that are increasing in time? Decreasing?

y'(t) = cos y for |y| ≤ π

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Textbook Question

{Use of Tech} Free fall An object in free fall may be modeled by assuming the only forces at work are the gravitational force and air resistance. By Newton’s Second Law of Motion (mass end . acceleration = the sum of external forces), the velocity of the object satisfies the differential equation

m · v'(t) = mg + f(v)

mass | acceleration | external forces

where f is a function that models the air resistance (assuming the positive direction is downward). One common assumption (often used for motion in air) is that f(v)=−kv^2, for t≥0, where k>0 is a drag coefficient.

c. Find the solution of this separable equation assuming v(0)=0 and 0<v²<g/a.

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Textbook Question

Direction field analysis Consider the first-order initial value problem y'(t)=ay+b,y(0)=A for t≥0 where a, b, and A are real numbers.

c. Draw a representative direction field in the case that a<0. Show that if A>−b/a, then the solution decreases for t≥0, and that if A<−b/a, then the solution increases for t≥0.

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. The general solution of the equation yy'(x) = xe⁻ʸ can be found using integration by parts.

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