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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

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.


where P(t) is the population, for t ≥ 0, and r > 0 and K > 0 are given constants.


d. Find lim(t→∞) P(t) and check that the result is consistent with the graph in part (c).

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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? 

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Textbook Question

{Use of Tech} Free fall Using th e background given in Exercise 47, assume the resistance is given by f(v)=−Rv, for t≥0, where R>0 is a drag coefficient (an assumption often made for a heavy medium such as water or oil).


c. Find the solution of this separable equation assuming v(0)=0 and 0<v<g/b.

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Textbook Question

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


c. Which time step results in the more accurate approximation? Explain your observations.


y′(t) = 4−y, y(0) = 3; y(t) = 4−e⁻ᵗ

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Textbook Question

33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.

d. Compare the errors in the approximations to y(T).


y′(t) = 6 - 2y, y(0) = -1; Δt = 0.2, T = 3; y(t) = 3 - 4e⁻²ᵗ

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Textbook Question

A second-order equation Consider the differential equation y''(t) - k²y(t) = 0 where k > 0 is a real number.


d. For a positive real number k, verify that the general solution of the equation may also be expressed in the form y(t) = C₁cosh(kt) + C₂sinh(kt), where cosh and sinh are the hyperbolic cosine and hyperbolic sine, respectively (Section 7.3).

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