Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. The solution of a stirred tank initial value problem always approaches a constant as t→∞
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Ch. 9 - Differential Equations
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
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 guidance1
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.
Recommended video:
07:39
Classifying Differential Equations
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.
Recommended video:
04:00Solutions to Basic Differential Equations
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.
Recommended video:
05:45Understanding Slope Fields
Related Practice
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
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Textbook Question
{Use of Tech} Tumor growth The Gompertz growth equation is often used to model the growth of tumors. Let M(t) be the mass of a tumor at time t≥0. The relevant initial value problem is
dM/dt=−rM ln(M/K),M(0)=M0,
where r and K are positive constants and 0<M0<K.
b. Solve the initial value problem and graph the solution for r=1,K=4, and M0=1. Describe the growth pattern of the tumor. Is the growth unbounded? If not, what is the limiting size of the tumor?
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Textbook Question
23–26. Stirred tank reactions For each of the following stirred tank reactions, carry out the following analysis.
b. Solve the initial value problem.
A one-million-liter pond is contaminated by a chemical pollutant with a concentration of 20 g/L. The source of the pollutant is removed, and pure water is allowed to flow into the pond at a rate of 1200 L/hr. Assuming the pond is thoroughly mixed and drained at a rate of 1200 L/hr, how long does it take to reduce the concentration of the solution in the pond to 10% of the initial value?
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Textbook Question
17–20. Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed.
b. In what regions are solutions increasing? Decreasing?
y'(t) = y(y+3)(4-y)
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