In each of Problems 1 through 6 draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as t . If this behavior depends on the initial value of y at t = 0, describe the dependency.y = 3 2y
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Table of Contents
1.1
Some Basic Mathematical Models; Direction Fields
1.2
Solutions of Some Differential Equations
1.3
Classification of Differential Equations
2
First Order Difference Equations
2.1
Linear Equations; Method of Integrating Factors
2.2
Separable Equations
2.3
Modeling with First Order Equations
2.4
Differences Between Linear and Nonlinear Equations
2.5
Autonomous Equations and Population Dynamics
2.6
Exact Equations and Integrating Factors
2.7
Numerical Approximations: Eulers Method
2.8
The Existence and Uniqueness Theorem
2.9
First Order Difference Equations
3.1
Homogeneous Equations with Constant Coefficients
3.2
Solutions of Linear Homogeneous Equations; the Wronskian
3.3
Complex Roots of the Characteristic Equation
3.4
Repeated Roots; Reduction of Order
3.5
Nonhomogeneous Equations; Method of Undetermined Coefficients
3.6
Variation of Parameters
3.7
Mechanical and Electrical Vibrations
3.8
Forced Vibrations
4.1
General Theory of nth Order Linear Equations
4.2
Homogeneous Equations with Constant Coefficients
4.3
The Method of Undetermined Coefficients
4.4
The Method of Variation of Parameters
5.1
Review of Power Series
5.2
Series Solutions Near an Ordinary Point, Part I
5.3
Series Solutions Near an Ordinary Point, Part II
5.4
Euler Equations; Regular Singular Points
5.5
Series Solutions Near a Regular Singular Point, Part I
5.6
Series Solutions Near a Regular Singular Point, Part II
5.7
Bessels Equation
6.1
Definition of the Laplace Transform
6.2
Solution of Initial Value Problems
6.3
Step Functions
6.4
Differential Equations with Discontinuous Forcing Functions
6.5
Impulse Functions
6.6
The Convolution Integral
7.1
Introduction
7.2
Review of Matrices
7.3
Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors
7.4
Basic Theory of Systems of First Order Linear Equations
7.5
Homogeneous Linear Systems with Constant Coefficients
7.6
Complex Eigenvalues
7.7
Fundamental Matrices
7.8
Repeated Eigenvalues
7.9
Nonhomogeneous Linear Systems
8.1
The Euler or Tangent Line Method
8.2
Improvements on the Euler Method
8.3
The RungeKutta Method
8.4
Multistep Methods
8.5
More on Errors; Stability
8.6
Systems of First Order Equations
9.1
The Phase Plane: Linear Systems
9.2
Autonomous Systems and Stability
9.3
Locally Linear Systems
9.4
Competing Species
9.5
PredatorPrey Equations
9.6
Liapunovs Second Method
9.7
Periodic Solutions and Limit Cycles
9.8
Chaos and Strange Attractors: The Lorenz Equations
10.1
Two-Point Boundary Value Problems
10.2
Fourier Series
10.3
The Fourier Convergence Theorem
10.4
Even and Odd Functions
10.5
Separation of Variables; Heat Conduction in a Rod
10.6
Other Heat Conduction Problems
10.7
The Wave Equation: Vibrations of an Elastic String
10.8
Laplaces Equation
11.1
The Occurrence of Two-Point Boundary Value Problems
11.2
SturmLiouville Boundary Value Problems
11.3
Nonhomogeneous Boundary Value Problems
11.4
Singular SturmLiouville Problems
11.5
Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion
11.6
Series of Orthogonal Functions: Mean Convergence
Textbook Solutions for Elementary Differential Equations and Boundary Value Problems
Chapter 1.1 Problem 11
Question
In each of 11 through 14 draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as t . If this behavior depends on the initial value of y at t = 0, describe this dependency. Note that in these problems the equations are not of the form y = ay + b, and the behavior of their solutions is somewhat more complicated than for the equations in the text.y = y(4 y)
Solution
The first step in solving 1.1 problem number 11 trying to solve the problem we have to refer to the textbook question: In each of 11 through 14 draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as t . If this behavior depends on the initial value of y at t = 0, describe this dependency. Note that in these problems the equations are not of the form y = ay + b, and the behavior of their solutions is somewhat more complicated than for the equations in the text.y = y(4 y)
From the textbook chapter Some Basic Mathematical Models; Direction Fields you will find a few key concepts needed to solve this.
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full solution
Title
Elementary Differential Equations and Boundary Value Problems 9
Author
Boyce, Richard C. DiPrima
ISBN
9780470383346
Answer: In each of 11 through 14 draw a direction field for the given differential
Chapter 1.1 textbook questions
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Chapter 1: Problem 1 Elementary Differential Equations and Boundary Value Problems 9
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Chapter 1: Problem 2 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 6 draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as t . If this behavior depends on the initial value of y at t = 0, describe the dependency.y = 2y 3
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Chapter 1: Problem 3 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 6 draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as t . If this behavior depends on the initial value of y at t = 0, describe the dependency.y = 3 + 2y
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Chapter 1: Problem 4 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 6 draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as t . If this behavior depends on the initial value of y at t = 0, describe the dependency.y = 1 2y
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Chapter 1: Problem 5 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 6 draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as t . If this behavior depends on the initial value of y at t = 0, describe the dependency.y = 1 + 2y
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Chapter 1: Problem 6 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 1 through 6 draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as t . If this behavior depends on the initial value of y at t = 0, describe the dependency.y = y + 2
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Chapter 1: Problem 7 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 7 through 10 write down a differential equation of the form dy/dt = ay + b whose solutions have the required behavior as t .All solutions approach y = 3.
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Chapter 1: Problem 8 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 7 through 10 write down a differential equation of the form dy/dt = ay + b whose solutions have the required behavior as t .All solutions approach y = 2/3
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Chapter 1: Problem 9 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 7 through 10 write down a differential equation of the form dy/dt = ay + b whose solutions have the required behavior as t .
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Chapter 1: Problem 10 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 7 through 10 write down a differential equation of the form dy/dt = ay + b whose solutions have the required behavior as t .All other solutions diverge from y = 2.All other solutions diverge from y = 1/3.
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Chapter 1: Problem 11 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 11 through 14 draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as t . If this behavior depends on the initial value of y at t = 0, describe this dependency. Note that in these problems the equations are not of the form y = ay + b, and the behavior of their solutions is somewhat more complicated than for the equations in the text.y = y(4 y)
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Chapter 1: Problem 12 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 11 through 14 draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as t . If this behavior depends on the initial value of y at t = 0, describe this dependency. Note that in these problems the equations are not of the form y = ay + b, and the behavior of their solutions is somewhat more complicated than for the equations in the text.
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Chapter 1: Problem 13 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 11 through 14 draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as t . If this behavior depends on the initial value of y at t = 0, describe this dependency. Note that in these problems the equations are not of the form y = ay + b, and the behavior of their solutions is somewhat more complicated than for the equations in the text.
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Chapter 1: Problem 14 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 11 through 14 draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as t . If this behavior depends on the initial value of y at t = 0, describe this dependency. Note that in these problems the equations are not of the form y = ay + b, and the behavior of their solutions is somewhat more complicated than for the equations in the text.y = y(y 2)2
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Chapter 1: Problem 15 Elementary Differential Equations and Boundary Value Problems 9
Consider the following list of differential equations, some of which produced the direction fields shown in Figures 1.1.5 through 1.1.10. In each of Problems 15 through 20 identify the differential equation that corresponds to the given direction field.(a) y = 2y 1 (b) y = 2 + y (c) y = y 2 (d) y = y(y + 3) (e) y = y(y 3) (f) y = 1 + 2y (g) y = 2 y (h) y = y(3 y) (i) y = 1 2y (j) y = 2 y
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Chapter 1: Problem 16 Elementary Differential Equations and Boundary Value Problems 9
Consider the following list of differential equations, some of which produced the direction fields shown in Figures 1.1.5 through 1.1.10. In each of Problems 15 through 20 identify the differential equation that corresponds to the given direction field.(a) y = 2y 1 (b) y = 2 + y (c) y = y 2 (d) y = y(y + 3) (e) y = y(y 3) (f) y = 1 + 2y (g) y = 2 y (h) y = y(3 y) (i) y = 1 2y (j) y = 2 y
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Chapter 1: Problem 17 Elementary Differential Equations and Boundary Value Problems 9
Consider the following list of differential equations, some of which produced the direction fields shown in Figures 1.1.5 through 1.1.10. In each of Problems 15 through 20 identify the differential equation that corresponds to the given direction field.(a) y = 2y 1 (b) y = 2 + y (c) y = y 2 (d) y = y(y + 3) (e) y = y(y 3) (f) y = 1 + 2y (g) y = 2 y (h) y = y(3 y) (i) y = 1 2y (j) y = 2 y
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Chapter 1: Problem 18 Elementary Differential Equations and Boundary Value Problems 9
Consider the following list of differential equations, some of which produced the direction fields shown in Figures 1.1.5 through 1.1.10. In each of Problems 15 through 20 identify the differential equation that corresponds to the given direction field.(a) y = 2y 1 (b) y = 2 + y (c) y = y 2 (d) y = y(y + 3) (e) y = y(y 3) (f) y = 1 + 2y (g) y = 2 y (h) y = y(3 y) (i) y = 1 2y (j) y = 2 y
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Chapter 1: Problem 19 Elementary Differential Equations and Boundary Value Problems 9
Consider the following list of differential equations, some of which produced the direction fields shown in Figures 1.1.5 through 1.1.10. In each of Problems 15 through 20 identify the differential equation that corresponds to the given direction field.(a) y = 2y 1 (b) y = 2 + y (c) y = y 2 (d) y = y(y + 3) (e) y = y(y 3) (f) y = 1 + 2y (g) y = 2 y (h) y = y(3 y) (i) y = 1 2y (j) y = 2 y
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Chapter 1: Problem 20 Elementary Differential Equations and Boundary Value Problems 9
Consider the following list of differential equations, some of which produced the direction fields shown in Figures 1.1.5 through 1.1.10. In each of Problems 15 through 20 identify the differential equation that corresponds to the given direction field.(a) y = 2y 1 (b) y = 2 + y (c) y = y 2 (d) y = y(y + 3) (e) y = y(y 3) (f) y = 1 + 2y (g) y = 2 y (h) y = y(3 y) (i) y = 1 2y (j) y = 2 y
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Chapter 1: Problem 21 Elementary Differential Equations and Boundary Value Problems 9
A pond initially contains 1,000,000 gal of water and an unknown amount of an undesirable chemical. Water containing 0.01 g of this chemical per gallon flows into the pond at a rate of 300 gal/h. The mixture flows out at the same rate, so the amount of water in the pond remains constant. Assume that the chemical is uniformly distributed throughout the pond. (a) Write a differential equation for the amount of chemical in the pond at any time. (b) How much of the chemical will be in the pond after a very long time? Does this limiting amount depend on the amount that was present initially?
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Chapter 1: Problem 22 Elementary Differential Equations and Boundary Value Problems 9
A spherical raindrop evaporates at a rate proportional to its surface area. Write a differential equation for the volume of the raindrop as a function of time.
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Chapter 1: Problem 23 Elementary Differential Equations and Boundary Value Problems 9
Newtons law of cooling states that the temperature of an object changes at a rate proportional to the difference between the temperature of the object itself and the temperature of its surroundings (the ambient air temperature in most cases). Suppose that the ambient temperature is 70F and that the rate constant is 0.05 (min)1. Write a differential equation for the temperature of the object at any time. Note that the differential equation is the same whether the temperature of the object is above or below the ambient temperature.
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Chapter 1: Problem 24 Elementary Differential Equations and Boundary Value Problems 9
A certain drug is being administered intravenously to a hospital patient. Fluid containing 5 mg/cm3 of the drug enters the patients bloodstream at a rate of 100 cm3/h. The drug is absorbed by body tissues or otherwise leaves the bloodstream at a rate proportional to the amount present, with a rate constant of 0.4 (h)1. (a) Assuming that the drug is always uniformly distributed throughout the bloodstream, write a differential equation for the amount of the drug that is present in the bloodstream at any time. (b) How much of the drug is present in the bloodstream after a long time?
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Chapter 1: Problem 25 Elementary Differential Equations and Boundary Value Problems 9
For small, slowly falling objects, the assumption made in the text that the drag force is proportional to the velocity is a good one. For larger, more rapidly falling objects, it is more accurate to assume that the drag force is proportional to the square of the velocity.2 (a) Write a differential equation for the velocity of a falling object of mass m if the drag force is proportional to the square of the velocity. (b) Determine the limiting velocity after a long time.(c) If m = 10 kg, find the drag coefficient so that the limiting velocity is 49 m/s. (d) Using the data in part (c), draw a direction field and compare it with Figure 1.1.3.
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Chapter 1: Problem 26 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 26 through 33 draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as t . If this behavior depends on the initial value of y at t = 0, describe this dependency. Note that the right sides of these equations depend on t as well as y; therefore their solutions can exhibit more complicated behavior than those in the text.y = 2 + t y
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Chapter 1: Problem 27 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 26 through 33 draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as t . If this behavior depends on the initial value of y at t = 0, describe this dependency. Note that the right sides of these equations depend on t as well as y; therefore their solutions can exhibit more complicated behavior than those in the text.
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Chapter 1: Problem 28 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 26 through 33 draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as t . If this behavior depends on the initial value of y at t = 0, describe this dependency. Note that the right sides of these equations depend on t as well as y; therefore their solutions can exhibit more complicated behavior than those in the text.
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Chapter 1: Problem 29 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 26 through 33 draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as t . If this behavior depends on the initial value of y at t = 0, describe this dependency. Note that the right sides of these equations depend on t as well as y; therefore their solutions can exhibit more complicated behavior than those in the text.y = t + 2y
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Chapter 1: Problem 30 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 26 through 33 draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as t . If this behavior depends on the initial value of y at t = 0, describe this dependency. Note that the right sides of these equations depend on t as well as y; therefore their solutions can exhibit more complicated behavior than those in the text.y = 3 sin t + 1 + y
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Chapter 1: Problem 31 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 26 through 33 draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as t . If this behavior depends on the initial value of y at t = 0, describe this dependency. Note that the right sides of these equations depend on t as well as y; therefore their solutions can exhibit more complicated behavior than those in the text.y = 2t 1 y2
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Chapter 1: Problem 32 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 26 through 33 draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as t . If this behavior depends on the initial value of y at t = 0, describe this dependency. Note that the right sides of these equations depend on t as well as y; therefore their solutions can exhibit more complicated behavior than those in the text.
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Chapter 1: Problem 33 Elementary Differential Equations and Boundary Value Problems 9
In each of Problems 26 through 33 draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as t . If this behavior depends on the initial value of y at t = 0, describe this dependency. Note that the right sides of these equations depend on t as well as y; therefore their solutions can exhibit more complicated behavior than those in the text.y = 1 6 y3 y 1 3 t
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