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 9.2.46a

Chapter 9, Problem 9.2.46a

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.

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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.

Rewrite the differential equation in the standard form for direction fields by isolating \(m'(t)\): \(m'(t) = I - k m(t)\) which becomes \(m'(t) = 10 - 0.05 m(t)\).

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 \(10 - 0.05 m\).

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.

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.

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Key Concepts

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

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.

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Textbook Question

{Use of Tech} Logistic equation for spread of rumors Sociologists model the spread of rumors using logistic equations. The key assumption is that at any given time, a fraction y of the population, where 0≤y≤1, knows the rumor, while the remaining fraction 1−y does not. Furthermore, the rumor spreads by interactions between those who know the rumor and those who do not. The number of such interactions is proportional to y1−y. Therefore, the equation that describes the spread of the rumor is y′(t)=ky(1−y), for t≥0 where k is a positive real number and t is measured in weeks. The number of people who initially know the rumor is y(0)=y0, where 0≤y0≤1.

a. Solve this initial value problem and give the solution in terms of k and y0.

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Textbook Question

{Use of Tech} Analytical solution of the predator-prey equations The solution of the predator-prey equations

X'(t) = -ax + bxy,y’(t) = cy - dxy

can be viewed as parametric equations that describe the solution curves. Assume a, b, c, and d are positive constants and consider solutions in the first quadrant.

a. Recalling that dy/dx = y(t)/x′(t), divide the first equation by the second equation to obtain a separable differential equation in terms of x and y.

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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.

a. Explain why y=−b/a is an equilibrium solution and corresponds to a horizontal line in the direction field.

Textbook Question

38–43. Equilibrium solutions A differential equation of the form y′(t)=f(y) is said to be autonomous (the function f depends only on y). The constant function y=y0 is an equilibrium solution of the equation provided f(y0)=0 (because then y'(t)=0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.

a. Find the equilibrium solutions.

y′(t) = y(2 - y)

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Textbook Question

a. Find the equilibrium solutions.

y′(t) = 2y + 4

Textbook Question

Consider the differential equation y'(t)+9y(t)=10.

a. How many arbitrary constants appear in the general solution of the differential equation?

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