Skip to main content
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 9.2.20b
Chapter 9, Problem 9.2.20b

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)

Verified step by step guidance
1
Identify the differential equation given: \(y'(t) = y(y+3)(4-y)\).
Determine the critical points by setting the right-hand side equal to zero: solve \(y(y+3)(4-y) = 0\). The solutions are \(y = 0\), \(y = -3\), and \(y = 4\).
Divide the real line into intervals based on these critical points: \((-\infty, -3)\), \((-3, 0)\), \((0, 4)\), and \((4, \infty)\).
Analyze the sign of \(y'(t)\) in each interval by choosing a test value from each interval and substituting it into \(y(y+3)(4-y)\) to determine if the product is positive or negative.
Conclude that solutions are increasing where \(y'(t) > 0\) and decreasing where \(y'(t) < 0\) based on the sign analysis in each interval.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

Key Concepts

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

Sign of the Derivative and Monotonicity

The sign of the derivative y'(t) determines whether the solution y(t) is increasing or decreasing. If y'(t) > 0, the function is increasing; if y'(t) < 0, it is decreasing. Analyzing the sign of y'(t) over intervals helps identify where solutions rise or fall.
Recommended video:
05:44
Derivatives

Factoring and Critical Points

Factoring the derivative expression y'(t) = y(y+3)(4-y) reveals critical points where y'(t) = 0, specifically at y = 0, y = -3, and y = 4. These points divide the y-axis into intervals where the sign of y'(t) can be tested to determine increasing or decreasing behavior.
Recommended video:
04:50
Critical Points

Interval Testing for Sign Analysis

To determine where y'(t) is positive or negative, select test values from each interval defined by the critical points. Substituting these values into y'(t) shows the sign of the derivative, which indicates whether the solution is increasing or decreasing in that region.
Recommended video:
07:09
The First Derivative Test: Finding Local Extrema
Related Practice
Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. The general solution of the separable equation y'(t) = t/(y' + 10y⁴) can be expressed explicitly with y in terms of t.

1
views
Textbook Question

52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.


{Use of Tech} Tumor growth The growth of cancer tumors may be modeled by the Gompertz growth equation. Let M(t) be the mass of a tumor, for t ≥ 0. The relevant initial value problem is:


dM/dt = -rM(t)ln(M(t)/K), M(0) = M₀,


where r and K are positive constants and 0 < M₀ < K.


b. Graph the solution for M₀ = 100 and r = 0.05.

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 500-L tank is initially filled with pure water. A copper sulfate solution with a concentration of 20 g/L flows into the tank at a rate of 4 L/min. The thoroughly mixed solution is drained from the tank at a rate of 4 L/min.

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 − 4xy, y′(t) = −y + 2xy

1
views
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?

Textbook Question

Euler’s method on more general grids Suppose the solution of the initial value problem y'(t)=f(t,y),y(a)=A is to be approximated on the interval [a, b].

b. Write the first step of Euler’s method to compute u1.