A first course in differential equations with modeling applications

NINTH EDITION

A FIRST COURSE IN

DIFFERENTIAL
EQUATIONS
with Modeling Applications


This page intentionally left blank


NINTH EDITION

A FIRST COURSE IN

DIFFERENTIAL
EQUATIONS
with Modeling Applications

DENNIS G. ZILL
Loyola Marymount University

Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States


A First Course in Differential
Equations with Modeling
Applications, Ninth Edition
Dennis G. Zill

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1 2 3 4 5 6 7 12 11 10 09 08


CONTENTS
Preface

1

ix

INTRODUCTION TO DIFFERENTIAL EQUATIONS
1.1 Definitions and Terminology
1.2 Initial-Value Problems

1

2

13

1.3 Differential Equations as Mathematical Models
CHAPTER 1 IN REVIEW

2

32

FIRST-ORDER DIFFERENTIAL EQUATIONS

34

2.1 Solution Curves Without a Solution
2.1.1

Direction Fields

2.1.2

Autonomous First-Order DEs

2.2 Separable Variables
2.3 Linear Equations

35

35
37

44
53

2.4 Exact Equations

62

2.5 Solutions by Substitutions
2.6 A Numerical Method
CHAPTER 2 IN REVIEW

3

19

70

75
80

MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS
3.1 Linear Models

82

83

3.2 Nonlinear Models

94

3.3 Modeling with Systems of First-Order DEs
CHAPTER 3 IN REVIEW

105

113

v


vi

4

●

CONTENTS

HIGHER-ORDER DIFFERENTIAL EQUATIONS

117

4.1 Preliminary Theory—Linear Equations

118

4.1.1

Initial-Value and Boundary-Value Problems

4.1.2

Homogeneous Equations

4.1.3

Nonhomogeneous Equations

4.2 Reduction of Order

118

120
125

130

4.3 Homogeneous Linear Equations with Constant Coefficients
4.4 Undetermined Coefficients—Superposition Approach
4.5 Undetermined Coefficients—Annihilator Approach
4.6 Variation of Parameters

157

4.7 Cauchy-Euler Equation

162

4.8 Solving Systems of Linear DEs by Elimination
4.9 Nonlinear Differential Equations
CHAPTER 4 IN REVIEW

5

140
150

169

174

178

MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS
5.1 Linear Models: Initial-Value Problems
5.1.1

Spring/Mass Systems: Free Undamped Motion

5.1.2

Spring/Mass Systems: Free Damped Motion

5.1.3

Spring/Mass Systems: Driven Motion

5.1.4

Series Circuit Analogue

5.3 Nonlinear Models

207

CHAPTER 5 IN REVIEW

216

6.1.1

Review of Power Series

6.1.2

Power Series Solutions

223
231

241

6.3.2

Legendre’s Equation

CHAPTER 6 IN REVIEW

220
220

6.2 Solutions About Singular Points
Bessel’s Equation

253

186

189
199

219

6.1 Solutions About Ordinary Points

6.3.1

182

192

SERIES SOLUTIONS OF LINEAR EQUATIONS

6.3 Special Functions

181

182

5.2 Linear Models: Boundary-Value Problems

6

133

241

248


CONTENTS

7

THE LAPLACE TRANSFORM
256

7.2 Inverse Transforms and Transforms of Derivatives
7.2.1

Inverse Transforms

7.2.2

Transforms of Derivatives

7.3 Operational Properties I

265

270

7.3.1

Translation on the s-Axis

271

7.3.2

Translation on the t-Axis

274

282

7.4.1

Derivatives of a Transform

7.4.2

Transforms of Integrals

7.4.3

Transform of a Periodic Function

7.5 The Dirac Delta Function

282
283
287

292

7.6 Systems of Linear Differential Equations

CHAPTER 7 IN REVIEW

262

262

7.4 Operational Properties II

295

300

SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS
8.1 Preliminary Theory—Linear Systems
8.2 Homogeneous Linear Systems
8.2.1

Distinct Real Eigenvalues

8.2.2

Repeated Eigenvalues

315

8.2.3

Complex Eigenvalues

320

8.3.1

Undetermined Coefficients

8.3.2

Variation of Parameters

8.4 Matrix Exponential

334

CHAPTER 8 IN REVIEW

337

304
312

326
326
329

NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS
9.1 Euler Methods and Error Analysis
9.2 Runge-Kutta Methods
9.3 Multistep Methods

340

345
350

9.4 Higher-Order Equations and Systems
9.5 Second-Order Boundary-Value Problems
CHAPTER 9 IN REVIEW

362

303

311

8.3 Nonhomogeneous Linear Systems

9

vii

255

7.1 Definition of the Laplace Transform

8

●

353
358

339


viii

●

CONTENTS

APPENDICES
I

Gamma Function

II

Matrices

III

Laplace Transforms

APP-1

APP-3
APP-21

Answers for Selected Odd-Numbered Problems
Index

I-1

ANS-1


PREFACE
TO THE STUDENT
Authors of books live with the hope that someone actually reads them. Contrary to
what you might believe, almost everything in a typical college-level mathematics text
is written for you and not the instructor. True, the topics covered in the text are chosen to appeal to instructors because they make the decision on whether to use it in
their classes, but everything written in it is aimed directly at you the student. So I
want to encourage you—no, actually I want to tell you—to read this textbook! But
do not read this text like you would a novel; you should not read it fast and you
should not skip anything. Think of it as a workbook. By this I mean that mathematics should always be read with pencil and paper at the ready because, most likely, you
will have to work your way through the examples and the discussion. Read—oops,
work—all the examples in a section before attempting any of the exercises; the examples are constructed to illustrate what I consider the most important aspects of the
section, and therefore, reflect the procedures necessary to work most of the problems
in the exercise sets. I tell my students when reading an example, cover up the solution; try working it first, compare your work against the solution given, and then
resolve any differences. I have tried to include most of the important steps in each
example, but if something is not clear you should always try—and here is where
the pencil and paper come in again—to fill in the details or missing steps. This may
not be easy, but that is part of the learning process. The accumulation of facts followed by the slow assimilation of understanding simply cannot be achieved without
a struggle.
Specifically for you, a Student Resource and Solutions Manual (SRSM) is available as an optional supplement. In addition to containing solutions of selected problems from the exercises sets, the SRSM has hints for solving problems, extra examples, and a review of those areas of algebra and calculus that I feel are particularly
important to the successful study of differential equations. Bear in mind you do not
have to purchase the SRSM; by following my pointers given at the beginning of most
sections, you can review the appropriate mathematics from your old precalculus or
calculus texts.
In conclusion, I wish you good luck and success. I hope you enjoy the text and

the course you are about to embark on—as an undergraduate math major it was one
of my favorites because I liked mathematics that connected with the physical world.
If you have any comments, or if you find any errors as you read/work your way
through the text, or if you come up with a good idea for improving either it or the
SRSM, please feel free to either contact me or my editor at Brooks/Cole Publishing
Company:


TO THE INSTRUCTOR
WHAT IS NEW IN THIS EDITION?
First, let me say what has not changed. The chapter lineup by topics, the number and
order of sections within a chapter, and the basic underlying philosophy remain the
same as in the previous editions.

ix


x

●

PREFACE

In case you are examining this text for the first time, A First Course in
Differential Equations with Modeling Applications, 9th Edition, is intended for
either a one-semester or a one-quarter course in ordinary differential equations. The
longer version of the text, Differential Equations with Boundary-Value Problems,
7th Edition, can be used for either a one-semester course, or a two-semester course
covering ordinary and partial differential equations. This longer text includes six
more chapters that cover plane autonomous systems and stability, Fourier series and

Fourier transforms, linear partial differential equations and boundary-value problems, and numerical methods for partial differential equations. For a one semester
course, I assume that the students have successfully completed at least two semesters of calculus. Since you are reading this, undoubtedly you have already examined
the table of contents for the topics that are covered. You will not find a “suggested
syllabus” in this preface; I will not pretend to be so wise as to tell other teachers
what to teach. I feel that there is plenty of material here to pick from and to form a
course to your liking. The text strikes a reasonable balance between the analytical,
qualitative, and quantitative approaches to the study of differential equations. As far
as my “underlying philosophy” it is this: An undergraduate text should be written
with the student’s understanding kept firmly in mind, which means to me that the
material should be presented in a straightforward, readable, and helpful manner,
while keeping the level of theory consistent with the notion of a “first course.”
For those who are familiar with the previous editions, I would like to mention a
few of the improvements made in this edition.
• Contributed Problems Selected exercise sets conclude with one or two contributed problems. These problems were class-tested and submitted by instructors of differential equations courses and reflect how they supplement
their classroom presentations with additional projects.
• Exercises Many exercise sets have been updated by the addition of new problems to better test and challenge the students. In like manner, some exercise
sets have been improved by sending some problems into early retirement.
• Design This edition has been upgraded to a four-color design, which adds
depth of meaning to all of the graphics and emphasis to highlighted phrases.
I oversaw the creation of each piece of art to ensure that it is as mathematically correct as the text.
• New Figure Numeration It took many editions to do so, but I finally became
convinced that the old numeration of figures, theorems, and definitions had to
be changed. In this revision I have utilized a double-decimal numeration system. By way of illustration, in the last edition Figure 7.52 only indicates that
it is the 52nd figure in Chapter 7. In this edition, the same figure is renumbered
as Figure 7.6.5, where
Chapter Section

bb

7.6.5 ; Fifth figure in the section

I feel that this system provides a clearer indication to where things are, without the necessity of adding a cumbersome page number.
• Projects from Previous Editions Selected projects and essays from past
editions of the textbook can now be found on the companion website at
academic.cengage.com/math/zill.
STUDENT RESOURCES
• Student Resource and Solutions Manual, by Warren S. Wright, Dennis G. Zill,
and Carol D. Wright (ISBN 0495385662 (accompanies A First Course in
Differential Equations with Modeling Applications, 9e), 0495383163 (accompanies Differential Equations with Boundary-Value Problems, 7e)) provides reviews of important material from algebra and calculus, the solution of
every third problem in each exercise set (with the exception of the Discussion


PREFACE

●

xi

Problems and Computer Lab Assignments), relevant command syntax for the
computer algebra systems Mathematica and Maple, lists of important concepts, as well as helpful hints on how to start certain problems.
• DE Tools is a suite of simulations that provide an interactive, visual exploration of the concepts presented in this text. Visit academic.cengage.com/
math/zill to find out more or contact your local sales representative to ask
about options for bundling DE Tools with this textbook.
INSTRUCTOR RESOURCES
• Complete Solutions Manual, by Warren S. Wright and Carol D. Wright (ISBN
049538609X), provides worked-out solutions to all problems in the text.
• Test Bank, by Gilbert Lewis (ISBN 0495386065) Contains multiple-choice
and short-answer test items that key directly to the text.

ACKNOWLEDGMENTS

Compiling a mathematics textbook such as this and making sure that its thousands of
symbols and hundreds of equations are (mostly) accurate is an enormous task, but
since I am called “the author” that is my job and responsibility. But many people
besides myself have expended enormous amounts of time and energy in working
towards its eventual publication. So I would like to take this opportunity to express my
sincerest appreciation to everyone—most of them unknown to me—at Brooks/Cole
Publishing Company, at Cengage Learning, and at Hearthside Publication Services
who were involved in the publication of this new edition. I would, however, like to single out a few individuals for special recognition: At Brooks/Cole/Cengage, Cheryll
Linthicum, Production Project Manager, for her willingness to listen to an author’s
ideas and patiently answering the author’s many questions; Larry Didona for the
excellent cover designs; Diane Beasley for the interior design; Vernon Boes for supervising all the art and design; Charlie Van Wagner, sponsoring editor; Stacy Green for
coordinating all the supplements; Leslie Lahr, developmental editor, for her suggestions, support, and for obtaining and organizing the contributed problems; and at
Hearthside Production Services, Anne Seitz, production editor, who once again put all
the pieces of the puzzle together. Special thanks go to John Samons for the outstanding job he did reviewing the text and answer manuscript for accuracy.
I also extend my heartfelt appreciation to those individuals who took the time
out of their busy academic schedules to submit a contributed problem:
Ben Fitzpatrick, Loyola Marymount University
Layachi Hadji, University of Alabama
Michael Prophet, University of Northern Iowa
Doug Shaw, University of Northern Iowa
Warren S. Wright, Loyola Marymount University
David Zeigler, California State University—Sacramento
Finally, over the years these texts have been improved in a countless number of
ways through the suggestions and criticisms of the reviewers. Thus it is fitting to conclude with an acknowledgement of my debt to the following people for sharing their
expertise and experience.
REVIEWERS OF PAST EDITIONS
William Atherton, Cleveland State University
Philip Bacon, University of Florida
Bruce Bayly, University of Arizona
William H. Beyer, University of Akron

R.G. Bradshaw, Clarkson College


xii

●

PREFACE

Dean R. Brown, Youngstown State University
David Buchthal, University of Akron
Nguyen P. Cac, University of Iowa
T. Chow, California State University—Sacramento
Dominic P. Clemence, North Carolina Agricultural
and Technical State University
Pasquale Condo, University of Massachusetts—Lowell
Vincent Connolly, Worcester Polytechnic Institute
Philip S. Crooke, Vanderbilt University
Bruce E. Davis, St. Louis Community College at Florissant Valley
Paul W. Davis, Worcester Polytechnic Institute
Richard A. DiDio, La Salle University
James Draper, University of Florida
James M. Edmondson, Santa Barbara City College
John H. Ellison, Grove City College
Raymond Fabec, Louisiana State University
Donna Farrior, University of Tulsa
Robert E. Fennell, Clemson University
W.E. Fitzgibbon, University of Houston
Harvey J. Fletcher, Brigham Young University
Paul J. Gormley, Villanova

Terry Herdman, Virginia Polytechnic Institute and State University
Zdzislaw Jackiewicz, Arizona State University
S.K. Jain, Ohio University
Anthony J. John, Southeastern Massachusetts University
David C. Johnson, University of Kentucky—Lexington
Harry L. Johnson, V.P.I & S.U.
Kenneth R. Johnson, North Dakota State University
Joseph Kazimir, East Los Angeles College
J. Keener, University of Arizona
Steve B. Khlief, Tennessee Technological University (retired)
C.J. Knickerbocker, St. Lawrence University
Carlon A. Krantz, Kean College of New Jersey
Thomas G. Kudzma, University of Lowell
G.E. Latta, University of Virginia
Cecelia Laurie, University of Alabama
James R. McKinney, California Polytechnic State University
James L. Meek, University of Arkansas
Gary H. Meisters, University of Nebraska—Lincoln
Stephen J. Merrill, Marquette University
Vivien Miller, Mississippi State University
Gerald Mueller, Columbus State Community College
Philip S. Mulry, Colgate University
C.J. Neugebauer, Purdue University
Tyre A. Newton, Washington State University
Brian M. O’Connor, Tennessee Technological University
J.K. Oddson, University of California—Riverside
Carol S. O’Dell, Ohio Northern University
A. Peressini, University of Illinois, Urbana—Champaign
J. Perryman, University of Texas at Arlington
Joseph H. Phillips, Sacramento City College

Jacek Polewczak, California State University Northridge
Nancy J. Poxon, California State University—Sacramento
Robert Pruitt, San Jose State University
K. Rager, Metropolitan State College
F.B. Reis, Northeastern University
Brian Rodrigues, California State Polytechnic University


PREFACE

●

xiii

Tom Roe, South Dakota State University
Kimmo I. Rosenthal, Union College
Barbara Shabell, California Polytechnic State University
Seenith Sivasundaram, Embry–Riddle Aeronautical University
Don E. Soash, Hillsborough Community College
F.W. Stallard, Georgia Institute of Technology
Gregory Stein, The Cooper Union
M.B. Tamburro, Georgia Institute of Technology
Patrick Ward, Illinois Central College
Warren S. Wright, Loyola Marymount University
Jianping Zhu, University of Akron
Jan Zijlstra, Middle Tennessee State University
Jay Zimmerman, Towson University
REVIEWERS OF THE CURRENT EDITIONS

Layachi Hadji, University of Alabama

Ruben Hayrapetyan, Kettering University
Alexandra Kurepa, North Carolina A&T State University
Dennis G. Zill
Los Angeles


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NINTH EDITION

A FIRST COURSE IN

DIFFERENTIAL
EQUATIONS
with Modeling Applications


This page intentionally left blank


1

INTRODUCTION TO DIFFERENTIAL
EQUATIONS
1.1 Definitions and Terminology
1.2 Initial-Value Problems
1.3 Differential Equations as Mathematical Models
CHAPTER 1 IN REVIEW

The words differential and equations certainly suggest solving some kind of
equation that contains derivatives yЈ, yЉ, . . . . Analogous to a course in algebra and
trigonometry, in which a good amount of time is spent solving equations such as
x2 ϩ 5x ϩ 4 ϭ 0 for the unknown number x, in this course one of our tasks will be
to solve differential equations such as yЉ ϩ 2yЈ ϩ y ϭ 0 for an unknown function
y ϭ ␾(x).
The preceding paragraph tells something, but not the complete story, about the
course you are about to begin. As the course unfolds, you will see that there is more
to the study of differential equations than just mastering methods that someone has
devised to solve them.
But first things first. In order to read, study, and be conversant in a specialized
subject, you have to learn the terminology of that discipline. This is the thrust of the
first two sections of this chapter. In the last section we briefly examine the link
between differential equations and the real world. Practical questions such as How
fast does a disease spread? How fast does a population change? involve rates of
change or derivatives. As so the mathematical description—or mathematical
model —of experiments, observations, or theories may be a differential equation.

1


2

●

CHAPTER 1

1.1

INTRODUCTION TO DIFFERENTIAL EQUATIONS

DEFINITIONS AND TERMINOLOGY
REVIEW MATERIAL
● Definition of the derivative
● Rules of differentiation
● Derivative as a rate of change
● First derivative and increasing/decreasing
● Second derivative and concavity
INTRODUCTION The derivative dy͞dx of a function y ϭ ␾(x) is itself another function ␾Ј(x)
2
found by an appropriate rule. The function y ϭ e0.1x is differentiable on the interval (Ϫϱ, ϱ), and
2
0.1x 2
by the Chain Rule its derivative is dy>dx ϭ 0.2xe . If we replace e0.1x on the right-hand side of
the last equation by the symbol y, the derivative becomes
dy
ϭ 0.2xy.
dx

(1)

Now imagine that a friend of yours simply hands you equation (1) —you have no idea how it was
constructed —and asks, What is the function represented by the symbol y? You are now face to face
with one of the basic problems in this course:
How do you solve such an equation for the unknown function y ϭ ␾(x)?

A DEFINITION The equation that we made up in (1) is called a differential
equation. Before proceeding any further, let us consider a more precise definition of
this concept.
DEFINITION 1.1.1 Differential Equation

An equation containing the derivatives of one or more dependent variables,
with respect to one or more independent variables, is said to be a differential
equation (DE).

To talk about them, we shall classify differential equations by type, order, and
linearity.
CLASSIFICATION BY TYPE If an equation contains only ordinary derivatives of
one or more dependent variables with respect to a single independent variable it is
said to be an ordinary differential equation (ODE). For example,
A DE can contain more
than one dependent variable

b

dy
ϩ 5y ϭ ex,
dx

2

d y dy
ϩ 6y ϭ 0,
Ϫ
dx2 dx

and

b

dx dy

ϩ
ϭ 2x ϩ y
dt
dt

(2)

are ordinary differential equations. An equation involving partial derivatives of
one or more dependent variables of two or more independent variables is called a


1.1

DEFINITIONS AND TERMINOLOGY

●

3

partial differential equation (PDE). For example,
Ѩ2u Ѩ2u
ϩ
ϭ 0,
Ѩx2 Ѩy2

Ѩ2u Ѩ2u
Ѩu
ϭ 2 Ϫ2 ,
2
Ѩx

Ѩt
Ѩt

and

Ѩu
Ѩv
ϭϪ
Ѩy
Ѩx

(3)

are partial differential equations.*
Throughout this text ordinary derivatives will be written by using either the
Leibniz notation dy͞dx, d 2 y͞dx 2, d 3 y͞dx 3, . . . or the prime notation yЈ, yЉ, yٞ, . . . .
By using the latter notation, the first two differential equations in (2) can be written
a little more compactly as yЈ ϩ 5y ϭ e x and yЉ Ϫ yЈ ϩ 6y ϭ 0. Actually, the prime
notation is used to denote only the first three derivatives; the fourth derivative is
written y (4) instead of yЉЉ. In general, the nth derivative of y is written d n y͞dx n or y (n).
Although less convenient to write and to typeset, the Leibniz notation has an advantage over the prime notation in that it clearly displays both the dependent and
independent variables. For example, in the equation
unknown function
or dependent variable

d 2x
–––2 ϩ 16x ϭ 0
dt
independent variable

it is immediately seen that the symbol x now represents a dependent variable,
whereas the independent variable is t. You should also be aware that in physical
sciences and engineering, Newton’s dot notation (derogatively referred to by some
as the “flyspeck” notation) is sometimes used to denote derivatives with respect
to time t. Thus the differential equation d 2s͞dt 2 ϭ Ϫ32 becomes s¨ ϭ Ϫ32. Partial
derivatives are often denoted by a subscript notation indicating the independent variables. For example, with the subscript notation the second equation in
(3) becomes u xx ϭ u tt Ϫ 2u t.
CLASSIFICATION BY ORDER The order of a differential equation (either
ODE or PDE) is the order of the highest derivative in the equation. For example,
second order

first order

( )

d 2y
dy 3
––––2 ϩ 5 ––– Ϫ 4y ϭ e x
dx
dx
is a second-order ordinary differential equation. First-order ordinary differential
equations are occasionally written in differential form M(x, y) dx ϩ N(x, y) dy ϭ 0.
For example, if we assume that y denotes the dependent variable in
(y Ϫ x) dx ϩ 4x dy ϭ 0, then yЈ ϭ dy͞dx, so by dividing by the differential dx, we
get the alternative form 4xyЈ ϩ y ϭ x. See the Remarks at the end of this section.
In symbols we can express an nth-order ordinary differential equation in one
dependent variable by the general form
F(x, y, yЈ, . . . , y(n)) ϭ 0,

(4)

where F is a real-valued function of n ϩ 2 variables: x, y, yЈ, . . . , y (n). For both practical and theoretical reasons we shall also make the assumption hereafter that it is
possible to solve an ordinary differential equation in the form (4) uniquely for the
*

Except for this introductory section, only ordinary differential equations are considered in A First
Course in Differential Equations with Modeling Applications, Ninth Edition. In that text the
word equation and the abbreviation DE refer only to ODEs. Partial differential equations or PDEs
are considered in the expanded volume Differential Equations with Boundary-Value Problems,
Seventh Edition.


4

●

CHAPTER 1

INTRODUCTION TO DIFFERENTIAL EQUATIONS

highest derivative y (n) in terms of the remaining n ϩ 1 variables. The differential
equation
d ny
ϭ f (x, y, yЈ, . . . , y(nϪ1)),
dxn

(5)

where f is a real-valued continuous function, is referred to as the normal form of (4).
Thus when it suits our purposes, we shall use the normal forms

dy
ϭ f (x, y)
dx

and

d 2y
ϭ f (x, y, yЈ)
dx2

to represent general first- and second-order ordinary differential equations. For example,
the normal form of the first-order equation 4xyЈ ϩ y ϭ x is yЈ ϭ (x Ϫ y)͞4x; the normal
form of the second-order equation yЉ Ϫ yЈ ϩ 6y ϭ 0 is yЉ ϭ yЈ Ϫ 6y. See the Remarks.
CLASSIFICATION BY LINEARITY An nth-order ordinary differential equation (4)
is said to be linear if F is linear in y, yЈ, . . . , y (n). This means that an nth-order ODE is
linear when (4) is a n(x)y (n) ϩ a nϪ1(x)y (nϪ1) ϩ и и и ϩ a1(x)yЈ ϩ a 0 (x)y Ϫ g(x) ϭ 0 or
an(x)

dny
d nϪ1y
dy
ϩ a0(x)y ϭ g(x).
ϩ
a
(x)
ϩ и и и ϩ a1(x)
nϪ1
n
nϪ1
dx

dx
dx

(6)

Two important special cases of (6) are linear first-order (n ϭ 1) and linear secondorder (n ϭ 2) DEs:
a1(x)

dy
ϩ a0 (x)y ϭ g(x)
dx

and

a2 (x)

d 2y
dy
ϩ a0 (x)y ϭ g(x). (7)
ϩ a1(x)
dx2
dx

In the additive combination on the left-hand side of equation (6) we see that the characteristic two properties of a linear ODE are as follows:
• The dependent variable y and all its derivatives yЈ, yЉ, . . . , y (n) are of the
first degree, that is, the power of each term involving y is 1.
• The coefficients a 0, a1, . . . , a n of y, yЈ, . . . , y (n) depend at most on the
independent variable x.
The equations
(y Ϫ x)dx ϩ 4x dy ϭ 0,

yЉ Ϫ 2yЈ ϩ y ϭ 0,

and

d 3y
dy
ϩx
Ϫ 5y ϭ ex
dx3
dx

are, in turn, linear first-, second-, and third-order ordinary differential equations. We
have just demonstrated that the first equation is linear in the variable y by writing it in
the alternative form 4xyЈ ϩ y ϭ x. A nonlinear ordinary differential equation is simply one that is not linear. Nonlinear functions of the dependent variable or its derivatives, such as sin y or e yЈ, cannot appear in a linear equation. Therefore
nonlinear term:
coefficient depends on y

nonlinear term:
nonlinear function of y

(1 Ϫ y)yЈ ϩ 2y ϭ ex,

d 2y
––––2 ϩ sin y ϭ 0,
dx

nonlinear term:
power not 1

and

d 4y
––––4 ϩ y 2 ϭ 0
dx

are examples of nonlinear first-, second-, and fourth-order ordinary differential equations, respectively.
SOLUTIONS As was stated before, one of the goals in this course is to solve, or
find solutions of, differential equations. In the next definition we consider the concept of a solution of an ordinary differential equation.


1.1

DEFINITIONS AND TERMINOLOGY

●

5

DEFINITION 1.1.2 Solution of an ODE
Any function ␾, defined on an interval I and possessing at least n derivatives
that are continuous on I, which when substituted into an nth-order ordinary
differential equation reduces the equation to an identity, is said to be a
solution of the equation on the interval.

In other words, a solution of an nth-order ordinary differential equation (4) is a function ␾ that possesses at least n derivatives and for which
F(x, ␾ (x), ␾Ј(x), . . . , ␾ (n)(x)) ϭ 0

for all x in I.

We say that ␾ satisfies the differential equation on I. For our purposes we shall also
assume that a solution ␾ is a real-valued function. In our introductory discussion we
2
saw that y ϭ e0.1x is a solution of dy͞dx ϭ 0.2xy on the interval (Ϫϱ, ϱ).
Occasionally, it will be convenient to denote a solution by the alternative
symbol y(x).
INTERVAL OF DEFINITION You cannot think solution of an ordinary differential
equation without simultaneously thinking interval. The interval I in Definition 1.1.2
is variously called the interval of definition, the interval of existence, the interval
of validity, or the domain of the solution and can be an open interval (a, b), a closed
interval [a, b], an infinite interval (a, ϱ), and so on.

EXAMPLE 1

Verification of a Solution

Verify that the indicated function is a solution of the given differential equation on
the interval (Ϫϱ, ϱ).
(a) dy>dx ϭ xy1/2; y ϭ 161 x4

(b) yЉ Ϫ 2yЈ ϩ y ϭ 0; y ϭ xex

SOLUTION One way of verifying that the given function is a solution is to see, after
substituting, whether each side of the equation is the same for every x in the interval.

(a) From
left-hand side:

dy
1

1
ϭ
(4 ؒ x3) ϭ x3,
dx 16
4

right-hand side:

xy1/2 ϭ x ؒ

΂ ΃
1 4
x
16

1/2

ϭxؒ

΂14 x ΃ ϭ 41 x ,
2

3

we see that each side of the equation is the same for every real number x. Note
that y1/2 ϭ 14 x2 is, by definition, the nonnegative square root of 161 x4.
(b) From the derivatives yЈ ϭ xe x ϩ e x and yЉ ϭ xe x ϩ 2e x we have, for every real
number x,
left-hand side:

yЉ Ϫ 2yЈ ϩ y ϭ (xex ϩ 2ex ) Ϫ 2(xex ϩ ex ) ϩ xex ϭ 0,

right-hand side:

0.

Note, too, that in Example 1 each differential equation possesses the constant solution y ϭ 0, Ϫϱ Ͻ x Ͻ ϱ. A solution of a differential equation that is identically
zero on an interval I is said to be a trivial solution.
SOLUTION CURVE The graph of a solution ␾ of an ODE is called a solution
curve. Since ␾ is a differentiable function, it is continuous on its interval I of definition. Thus there may be a difference between the graph of the function ␾ and the


6

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CHAPTER 1

INTRODUCTION TO DIFFERENTIAL EQUATIONS

graph of the solution ␾. Put another way, the domain of the function ␾ need not be
the same as the interval I of definition (or domain) of the solution ␾. Example 2
illustrates the difference.

y

EXAMPLE 2

1
1

x

(a) function y ϭ 1/x, x

0

y

1
1

x

(b) solution y ϭ 1/x, (0, ȍ)

FIGURE 1.1.1 The function y ϭ 1͞x
is not the same as the solution y ϭ 1͞x

Function versus Solution

The domain of y ϭ 1͞x, considered simply as a function, is the set of all real numbers x except 0. When we graph y ϭ 1͞x, we plot points in the xy-plane corresponding to a judicious sampling of numbers taken from its domain. The rational
function y ϭ 1͞x is discontinuous at 0, and its graph, in a neighborhood of the origin, is given in Figure 1.1.1(a). The function y ϭ 1͞x is not differentiable at x ϭ 0,
since the y-axis (whose equation is x ϭ 0) is a vertical asymptote of the graph.
Now y ϭ 1͞x is also a solution of the linear first-order differential equation
xyЈ ϩ y ϭ 0. (Verify.) But when we say that y ϭ 1͞x is a solution of this DE, we
mean that it is a function defined on an interval I on which it is differentiable and
satisfies the equation. In other words, y ϭ 1͞x is a solution of the DE on any interval that does not contain 0, such as (Ϫ3, Ϫ1), ( 12, 10), (Ϫϱ, 0), or (0, ϱ). Because
the solution curves defined by y ϭ 1͞x for Ϫ3 Ͻ x Ͻ Ϫ1 and 12 Ͻ x Ͻ 10 are simply segments, or pieces, of the solution curves defined by y ϭ 1͞x for Ϫϱ Ͻ x Ͻ 0
and 0 Ͻ x Ͻ ϱ, respectively, it makes sense to take the interval I to be as large as

possible. Thus we take I to be either (Ϫϱ, 0) or (0, ϱ). The solution curve on (0, ϱ)
is shown in Figure 1.1.1(b).
EXPLICIT AND IMPLICIT SOLUTIONS You should be familiar with the terms
explicit functions and implicit functions from your study of calculus. A solution in
which the dependent variable is expressed solely in terms of the independent
variable and constants is said to be an explicit solution. For our purposes, let us
think of an explicit solution as an explicit formula y ϭ ␾(x) that we can manipulate,
evaluate, and differentiate using the standard rules. We have just seen in the last two
examples that y ϭ 161 x4, y ϭ xe x, and y ϭ 1͞x are, in turn, explicit solutions
of dy͞dx ϭ xy 1/2, yЉ Ϫ 2yЈ ϩ y ϭ 0, and xyЈ ϩ y ϭ 0. Moreover, the trivial solution y ϭ 0 is an explicit solution of all three equations. When we get down to
the business of actually solving some ordinary differential equations, you will
see that methods of solution do not always lead directly to an explicit solution
y ϭ ␾(x). This is particularly true when we attempt to solve nonlinear first-order
differential equations. Often we have to be content with a relation or expression
G(x, y) ϭ 0 that defines a solution ␾ implicitly.
DEFINITION 1.1.3 Implicit Solution of an ODE
A relation G(x, y) ϭ 0 is said to be an implicit solution of an ordinary
differential equation (4) on an interval I, provided that there exists at least
one function ␾ that satisfies the relation as well as the differential equation
on I.

It is beyond the scope of this course to investigate the conditions under which a
relation G(x, y) ϭ 0 defines a differentiable function ␾. So we shall assume that if
the formal implementation of a method of solution leads to a relation G(x, y) ϭ 0,
then there exists at least one function ␾ that satisfies both the relation (that is,
G(x, ␾(x)) ϭ 0) and the differential equation on an interval I. If the implicit solution
G(x, y) ϭ 0 is fairly simple, we may be able to solve for y in terms of x and obtain
one or more explicit solutions. See the Remarks.

1.1

y

5

DEFINITIONS AND TERMINOLOGY

●

7

EXAMPLE 3 Verification of an Implicit Solution
The relation x 2 ϩ y 2 ϭ 25 is an implicit solution of the differential equation

5
x

dy
x
ϭϪ
dx
y

(8)

on the open interval (Ϫ5, 5). By implicit differentiation we obtain
(a) implicit solution

d 2

d
d 2
x ϩ
y ϭ
25
dx
dx
dx

x 2 ϩ y 2 ϭ 25
y

5
x

y1 ϭ ͙25 Ϫ x 2, Ϫ 5 Ͻ x Ͻ 5
y
5

5
x

−5

(c) explicit solution
y2 ϭ Ϫ͙25 Ϫ x 2, Ϫ5 Ͻ x Ͻ 5

FIGURE 1.1.2 An implicit solution
and two explicit solutions of yЈ ϭ Ϫx͞y

y
c>0
c=0

xyЈ Ϫ y ϭ x 2 sin x

Some solutions of

dy
ϭ 0.
dx

Any relation of the form x 2 ϩ y 2 Ϫ c ϭ 0 formally satisfies (8) for any constant c.
However, it is understood that the relation should always make sense in the real number
system; thus, for example, if c ϭ Ϫ25, we cannot say that x 2 ϩ y 2 ϩ 25 ϭ 0 is an
implicit solution of the equation. (Why not?)
Because the distinction between an explicit solution and an implicit solution
should be intuitively clear, we will not belabor the issue by always saying, “Here is
an explicit (implicit) solution.”

(b) explicit solution

FIGURE 1.1.3

2x ϩ 2y

Solving the last equation for the symbol dy͞dx gives (8). Moreover, solving
x 2 ϩ y 2 ϭ 25 for y in terms of x yields y ϭ Ϯ 225 Ϫ x2. The two functions
y ϭ ␾1(x) ϭ 125 Ϫ x2 and y ϭ ␾2(x) ϭ Ϫ125 Ϫ x2 satisfy the relation (that is,
x 2 ϩ ␾12 ϭ 25 and x 2 ϩ ␾ 22 ϭ 25) and are explicit solutions defined on the interval

(Ϫ5, 5). The solution curves given in Figures 1.1.2(b) and 1.1.2(c) are segments of
the graph of the implicit solution in Figure 1.1.2(a).

5

c<0

or

x

FAMILIES OF SOLUTIONS The study of differential equations is similar to that of
integral calculus. In some texts a solution ␾ is sometimes referred to as an integral
of the equation, and its graph is called an integral curve. When evaluating an antiderivative or indefinite integral in calculus, we use a single constant c of integration.
Analogously, when solving a first-order differential equation F(x, y, yЈ) ϭ 0, we
usually obtain a solution containing a single arbitrary constant or parameter c. A
solution containing an arbitrary constant represents a set G(x, y, c) ϭ 0 of solutions
called a one-parameter family of solutions. When solving an nth-order differential
equation F(x, y, yЈ, . . . , y (n)) ϭ 0, we seek an n-parameter family of solutions
G(x, y, c1, c 2, . . . , cn ) ϭ 0. This means that a single differential equation can possess
an infinite number of solutions corresponding to the unlimited number of choices
for the parameter(s). A solution of a differential equation that is free of arbitrary
parameters is called a particular solution. For example, the one-parameter family
y ϭ cx Ϫ x cos x is an explicit solution of the linear first-order equation xyЈ Ϫ y ϭ
x 2 sin x on the interval (Ϫϱ, ϱ). (Verify.) Figure 1.1.3, obtained by using graphing software, shows the graphs of some of the solutions in this family. The solution y ϭ
Ϫx cos x, the blue curve in the figure, is a particular solution corresponding to c ϭ 0.
Similarly, on the interval (Ϫϱ, ϱ), y ϭ c1e x ϩ c 2 xe x is a two-parameter family of solutions of the linear second-order equation yЉ Ϫ 2yЈ ϩ y ϭ 0 in Example 1. (Verify.)
Some particular solutions of the equation are the trivial solution y ϭ 0 (c1 ϭ c 2 ϭ 0),
y ϭ xe x (c1 ϭ 0, c 2 ϭ 1), y ϭ 5e x Ϫ 2xe x (c1 ϭ 5, c2 ϭ Ϫ2), and so on.
Sometimes a differential equation possesses a solution that is not a member of a

family of solutions of the equation —that is, a solution that cannot be obtained by specializing any of the parameters in the family of solutions. Such an extra solution is called
a singular solution. For example, we have seen that y ϭ 161 x4 and y ϭ 0 are solutions of
the differential equation dy͞dx ϭ xy 1/2 on (Ϫϱ, ϱ). In Section 2.2 we shall demonstrate,
by actually solving it, that the differential equation dy͞dx ϭ xy 1/2 possesses the oneparameter family of solutions y ϭ (14 x2 ϩ c)2. When c ϭ 0, the resulting particular
solution is y ϭ 161 x4. But notice that the trivial solution y ϭ 0 is a singular solution, since


8

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CHAPTER 1

INTRODUCTION TO DIFFERENTIAL EQUATIONS

it is not a member of the family y ϭ (14 x2 ϩ c) 2; there is no way of assigning a value to
the constant c to obtain y ϭ 0.
In all the preceding examples we used x and y to denote the independent and
dependent variables, respectively. But you should become accustomed to seeing
and working with other symbols to denote these variables. For example, we could
denote the independent variable by t and the dependent variable by x.

EXAMPLE 4

Using Different Symbols

The functions x ϭ c1 cos 4t and x ϭ c 2 sin 4t, where c1 and c 2 are arbitrary constants
or parameters, are both solutions of the linear differential equation
xЉ ϩ 16x ϭ 0.
For x ϭ c1 cos 4t the first two derivatives with respect to t are xЈ ϭ Ϫ4c1 sin 4t

and xЉ ϭ Ϫ16c1 cos 4t. Substituting xЉ and x then gives
xЉ ϩ 16x ϭ Ϫ16c1 cos 4t ϩ 16(c1 cos 4t) ϭ 0.
In like manner, for x ϭ c 2 sin 4t we have xЉ ϭ Ϫ16c 2 sin 4t, and so
xЉ ϩ 16x ϭ Ϫ16c2 sin 4t ϩ 16(c2 sin 4t) ϭ 0.
Finally, it is straightforward to verify that the linear combination of solutions, or the
two-parameter family x ϭ c1 cos 4t ϩ c 2 sin 4t, is also a solution of the differential
equation.
The next example shows that a solution of a differential equation can be a
piecewise-defined function.

EXAMPLE 5

You should verify that the one-parameter family y ϭ cx 4 is a one-parameter family
of solutions of the differential equation xyЈ Ϫ 4y ϭ 0 on the inverval (Ϫϱ, ϱ). See
Figure 1.1.4(a). The piecewise-defined differentiable function

y
c=1
x
c = −1

y
c = 1,
x ≤0
x
c = −1,
x<0

(b) piecewise-defined solution

xyЈ Ϫ 4y ϭ 0

yϭ

ΆϪxx ,,
4
4

xϽ0
xՆ0

is a particular solution of the equation but cannot be obtained from the family
y ϭ cx 4 by a single choice of c; the solution is constructed from the family by choosing c ϭ Ϫ1 for x Ͻ 0 and c ϭ 1 for x Ն 0. See Figure 1.1.4(b).

(a) two explicit solutions

FIGURE 1.1.4

A Piecewise-Defined Solution

Some solutions of

SYSTEMS OF DIFFERENTIAL EQUATIONS Up to this point we have been
discussing single differential equations containing one unknown function. But
often in theory, as well as in many applications, we must deal with systems of
differential equations. A system of ordinary differential equations is two or more
equations involving the derivatives of two or more unknown functions of a single
independent variable. For example, if x and y denote dependent variables and t
denotes the independent variable, then a system of two first-order differential
equations is given by

dx
ϭ f(t, x, y)
dt
dy
ϭ g(t, x, y).
dt

(9)