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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 9.2.18d
Chapter 9, Problem 9.2.18d

17–20. Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed.


d. Sketch the direction field and verify that it is consistent with parts (a)–(c).


y'(t) = (y−2)(y+1)

Verified step by step guidance
1
Identify the differential equation given: \(y'(t) = (y - 2)(y + 1)\). This represents the rate of change of \(y\) with respect to \(t\) depending on the value of \(y\).
Find the equilibrium solutions by setting \(y'(t) = 0\). Solve \((y - 2)(y + 1) = 0\) to find the constant solutions where the slope is zero.
Analyze the sign of \(y'(t)\) in the intervals determined by the equilibrium points. For \(y < -1\), between \(-1\) and \(2\), and for \(y > 2\), determine whether \(y'(t)\) is positive or negative to understand where the solution is increasing or decreasing.
Sketch the direction field by drawing small line segments with slopes given by \(y'(t)\) at various points \((t, y)\). Since the equation depends only on \(y\), the slope at each horizontal line \(y = c\) is constant, making the direction field easier to visualize.
Verify consistency with parts (a)–(c) by checking that the increasing and decreasing behavior of solutions matches the sign analysis and the equilibrium points found. Confirm that solutions move away from unstable equilibria and toward stable equilibria as \(t\) increases.

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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 differential equation that shows the slope of the solution curve at various points. Each small line segment indicates the slope y' at that point (t, y), helping visualize the behavior of solutions without solving the equation explicitly.
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Understanding Slope Fields

Equilibrium Solutions

Equilibrium solutions occur where the derivative y' equals zero, meaning the function y(t) remains constant. For y' = (y−2)(y+1), the equilibria are y = 2 and y = -1, which are critical for understanding the long-term behavior of solutions and the shape of the direction field.
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Solutions to Basic Differential Equations

Increasing and Decreasing Solutions

The sign of y' determines whether solutions increase or decrease. If y' > 0, the solution is increasing; if y' < 0, it is decreasing. Analyzing the sign of (y−2)(y+1) in different intervals helps predict how solutions behave relative to equilibrium points.
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Determining Where a Function is Increasing & Decreasing
Related Practice
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.


e. Sketch a representative solution curve in the xy-plane and indicate the direction in which the solution evolves.


x′(t) = −3x + 6xy, y′(t) = y − 4xy

Textbook Question

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


d. According to Newton’s Law of Cooling, the temperature of a hot object will reach the ambient temperature after a finite amount of time.

Textbook Question

U.S. population projections According to the U.S. Census Bureau, the nation’s population (to the nearest million) was 296 million in 2005 and 321 million in 2015. The Bureau also projects a 2050 population of 398 million. To construct a logistic model, both the growth rate and the carrying capacity must be estimated. There are several ways to estimate these parameters. Here is one approach:


d. Estimations of this kind must be made and interpreted carefully. Suppose the projected population for 2050 is 410 million rather than 398 million. What is the value of the carrying capacity in this case?

Textbook Question

A physiological model A common assumption in modeling drug assimilation is that the blood volume in a person is a single compartment that behaves like a stirred tank. Suppose the blood volume is a four-liter tank that initially has a zero concentration of a particular drug. At time t = 0, an intravenous line is inserted into a vein (into the tank) that carries a drug solution with a concentration of 500 mg/L. The inflow rate is 0.06 L/min. Assume the drug is quickly mixed thoroughly in the blood and that the volume of blood remains constant.

d. After how many minutes does the drug mass reach 90% of its steady-state level? 

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.

d. Identify the four regions in the first quadrant of the xy-plane in which x' and y' are positive or negative.


x′(t) = 2x − 4xy, y′(t) = −y + 2xy