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Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 9 - Differential EquationsProblema 9.R.20d
Capitolo 9, Problema 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.

Guida verificata passo dopo passo
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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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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Percorso guidato
5:50
Asymptotes of Hyperbolas
Pratica correlata
Domanda del libro di testo

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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Domanda del libro di testo

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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Domanda del libro di testo

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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Domanda del libro di testo

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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Domanda del libro di testo

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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Domanda del libro di testo

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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