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Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 9 - Differential EquationsProblem 9.R.20d
Chapter 9, Problem 9.R.20d

Direction fields The direction field for the equation y′(t)=t−y, for |t|≤4 and |y|≤4, is shown in the figure.
d. Complete the following sentence. The solution of the differential equation with the initial condition y(0)=A, where A is a real number, approaches the line _____ as t→∞.
Direction field plot showing slope vectors for y′(t) = t − y over t and y from -4 to 4.

Verified step by step guidance
1
Identify the given differential equation: \(y'(t) = t - y\).
To find the behavior of solutions as \(t \to \infty\), consider the equilibrium or steady-state solution where \(y'(t) = 0\). Set \(0 = t - y\), which implies \(y = t\).
This suggests that as \(t\) becomes very large, the solution \(y(t)\) approaches the line \(y = t\).
To confirm this, rewrite the differential equation as \(y' + y = t\), which is a linear first-order ODE. The general solution will involve a particular solution plus a complementary solution that decays over time.
Since the complementary solution tends to zero as \(t \to \infty\), the solution \(y(t)\) approaches the particular solution \(y = t\), meaning the solution approaches the line \(y = t\) as \(t \to \infty\).

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Key Concepts

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

Direction Fields

A direction field is a graphical representation of a first-order differential equation showing slope vectors at various points. It helps visualize the behavior of solutions without solving the equation explicitly. Each small line segment indicates the slope y' at that (t, y) point, guiding the shape of solution curves.
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Understanding Slope Fields

Solving First-Order Linear Differential Equations

The equation y' = t - y is a first-order linear differential equation. Solutions can be found using integrating factors or recognizing it as a linear ODE. Understanding the general solution form helps predict long-term behavior and how initial conditions affect the solution.
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Solving Separable Differential Equations

Asymptotic Behavior and Equilibrium Solutions

As t approaches infinity, solutions to differential equations often approach a particular function or line called an asymptote or equilibrium solution. For y' = t - y, the solution tends to a line where the slope stabilizes, which can be found by setting y' = 0 or analyzing the steady-state behavior.
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Asymptotes of Hyperbolas
Related Practice
Textbook Question

Direction fields Consider the direction field for the equation y′=y(2−y) shown in the figure and initial conditions of the form y(0)=A.

a. Sketch a solution on the direction field with the initial condition y(0)=1.

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Textbook Question

Euler’s metho d Consider the initial value problem y′(t)=1/2y,y(0)=1. 

a. Use Euler’s method with Δt=0.1 to compute approximations to y(0.1) and y(0.2). 

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Textbook Question

Logistic growth The population of a rabbit community is governed by the initial value problem

P′(t) = 0.2 P (1 − P/1200), P(0) = 50

a. Find the equilibrium solutions.

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Textbook Question

A predator-prey model Consider the predator-prey model

x′(t) = −4x + 2xy, y′(t) = 5y − xy

c. Find the equilibrium points for the system.

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Textbook Question

Euler’s method Consider the initial value problem y′(t)=1/2y,y(0)=1. 

b. Use Euler’s method with Δt=0.05 to compute approximations to y(0.1) and y(0.2). 

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Textbook Question

Direction fields The direction field for the equation y′(t)=t−y, for |t|≤4 and |y|≤4, is shown in the figure.

b. Use the direction field to sketch the solution curve that passes through the point (0,−1/2).

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