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 9.2.39b

Chapter 9, Problem 9.2.39b

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.
b. Sketch the direction field, for t≥0.

y′(t) = 6 - 2y

Verified step by step guidance

Identify that the given differential equation is autonomous because the right-hand side depends only on y: \(y'(t) = 6 - 2y\).

Find the equilibrium solution(s) by setting \(y'(t) = 0\), which means solving \(6 - 2y = 0\) for \(y\).

Solve the equation \(6 - 2y = 0\) to find the equilibrium value(s) \(y_0\) where the slope of the solution is zero, indicating horizontal lines in the direction field.

To sketch the direction field for \(t \geq 0\), choose several values of \(y\) and compute the slope \(y'(t) = 6 - 2y\) at each of these points. Since the equation is autonomous, the slope depends only on \(y\), not on \(t\).

Draw short line segments at points \((t, y)\) with \(t \geq 0\) where the slope of each segment corresponds to the value of \(6 - 2y\). Include the equilibrium solution as a horizontal line where the slope is zero.

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Key Concepts

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

Autonomous Differential Equations

An autonomous differential equation is one where the derivative y' depends only on the variable y, not explicitly on the independent variable t. This means the rate of change of y depends solely on y's current value, simplifying analysis and allowing direction fields to be independent of t.

Equilibrium Solutions

Equilibrium solutions occur when y' = 0, meaning the function f(y) equals zero at some y = y0. These solutions are constant functions where the system remains steady over time, represented as horizontal lines in the direction field.

Direction Fields (Slope Fields)

A direction field is a graphical tool showing slopes y' = f(y) at various points (t, y). For autonomous equations, slopes depend only on y, so the field consists of horizontal slices with constant slopes, helping visualize solution behavior without solving the equation explicitly.

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Understanding Slope Fields

Related Practice

Textbook Question

29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.

b. Using the exact solution given, compute the errors in the Euler approximations at t=0.2 and t=0.4.

y′(t) = −y, y(0) = 1; y(t) = e⁻ᵗ

Textbook Question

42–43. Implicit solutions for separable equations For the following separable equations, carry out the indicated analysis.

b. Find the value of the arbitrary constant associated with each initial condition. (Each initial condition requires a different constant.)

y'(t) = t²/(y² + 1); y(−1) = 1, y(0) = 0, y(−1) = −1

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Textbook Question

Blowup in finite time Consider the initial value problem y'(t) = yⁿ + 1, y(0) = y₀, where n is a positive integer.

b. Solve the initial value problem with n = 2 and y₀ = 1/√2.

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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.

b. Sketch the direction field, for t≥0.

y′(t) = 2y + 4

Textbook Question

{Use of Tech} Chemical rate equations Let y(t) be t he concentration of a substance in a chemical reaction (typical units are moles/liter). The change in the concentration, under appropriate conditions, is modeled by the equation dy/dt=-ky^n for t≥0, where k>0 is a rate constant and the positive integer n is the order of the reaction.

b. Solve the initial value problem for a second-order reaction (n=2) assuming y(0)=y0.

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Textbook Question

27–30. Predator-prey models Consider the following pairs of differential equations that model a predator-prey system with populations x and y. In each case, carry out the following steps.

b. Find the lines along which x'(t) = 0. Find the lines along which y'(t) = 0.

x′(t) = 2x − xy, y′(t) = −y + xy

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