MATH261 West Virginia Linear Differential System Calculus Problems please solve the problems in the assignment paper screenshoted handwritten on a sheet of
MATH261 West Virginia Linear Differential System Calculus Problems please solve the problems in the assignment paper screenshoted handwritten on a sheet of paper with clear and neat steps and upload it here by taking a clear picture of it. I will be copying down the answers myself on a sheet of paper and submitting the work. please use the digital book attached as a reference to the number of questions and sections listed in the assignment paper. thank you very much. Boyce 9131 FM 2
October 12, 2016
17:16 iii
Elementary
Differential Equations and
Boundary Value Problems
Eleventh Edition
WILLIAM E. BOYCE
Edward P. Hamilton Professor Emeritus
Department of Mathematical Sciences
Rensselaer Polytechnic Institute
RICHARD C. DIPRIMA
formerly Eliza Ricketts Foundation Professor
Department of Mathematical Sciences
Rensselaer Polytechnic Institute
DOUGLAS B. MEADE
Department of Mathematics
University of South Carolina – Columbia
iii
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iv
Preface
As we have prepared an updated edition our first priorities
are to preserve, and to enhance, the qualities that have made
previous editions so successful. In particular, we adopt the
viewpoint of an applied mathematician with diverse interests
in differential equations, ranging from quite theoretical to
intensely practical–and usually a combination of both. Three
pillars of our presentation of the material are methods
of solution, analysis of solutions, and approximations of
solutions. Regardless of the specific viewpoint adopted, we
have sought to ensure the exposition is simultaneously correct
and complete, but not needlessly abstract.
The intended audience is undergraduate STEM students
whose degree program includes an introductory course in
differential equations during the first two years. The essential
prerequisite is a working knowledge of calculus, typically a
two- or three-semester course sequence or an equivalent. While
a basic familiarity with matrices is helpful, Sections 7.2 and 7.3
provide an overview of the essential linear algebra ideas needed
for the parts of the book that deal with systems of differential
equations (the remainder of Chapter 7, Section 8.5, and
Chapter 9).
A strength of this book is its appropriateness in a
wide variety of instructional settings. In particular, it allows
instructors flexibility in the selection of and the ordering of
topics and in the use of technology. The essential core material
is Chapter 1, Sections 2.1 through 2.5, and Sections 3.1 through
3.5. After completing these sections, the selection of additional
topics, and the order and depth of coverage are generally at
the discretion of the instructor. Chapters 4 through 11 are
essentially independent of each other, except that Chapter 7
should precede Chapter 9, and Chapter 10 should precede
Chapter 11.
A particularly appealing aspect of differential equations
is that even the simplest differential equations have a direct
correspondence to realistic physical phenomena: exponential
growth and decay, spring-mass systems, electrical circuits,
competitive species, and wave propagation. More complex
natural processes can often be understood by combining and
building upon simpler and more basic models. A thorough
knowledge of these basic models, the differential equations
that describe them, and their solutions–either explicit solutions
or qualitative properties of the solution–is the first and
indispensable step toward analyzing the solutions of more
complex and realistic problems. The modeling process is
detailed in Chapter 1 and Section 2.3. Careful constructions
of models appear also in Sections 2.5, 3.7, 9.4, 10.5, and
10.7 (and the appendices to Chap er 10). Various problem sets
throughout the book include problems that involve modeling
to formulate an appropriate differential equation, and then
to solve it or to determine some qualitative properties of its
solution. The primary purposes of these applied problems are
to provide students with hands-on experience in the derivation
of differential equations, and to convince them that differential
equations arise naturally in a wide variety of real-world
applications.
Another important concept emphasized repeatedly
throughout the book is the transportability of mathematical
knowledge. While a specific solution method applies to only a
particular class of differential equations, it can be used in any
application in which that particular type of differential equation
arises. Once this point is made in a convincing manner, we
believe that it is unnecessary to provide specific applications of
every method of solution or type of equation that we consider.
This decision helps to keep this book to a reasonable size, and
allows us to keep the primary emphasis on the development
of more solution methods for additional types of differential
equations.
From a student’s point of view, the problems that are
assigned as homework and that appear on examinations define
the course. We believe that the most outstanding feature of
this book is the number, and above all the variety and range,
of the problems that it contains. Many problems are entirely
straightforward, but many others are more challenging, and
some are fairly open-ended and can even serve as the basis
for independent student projects. The observant reader will
notice that there are fewer problems in this edition than in
previous editions; many of these problems remain available
to instructors via the WileyPlus course. The remaining 1600
problems are still far more problems than any instructor can
use in any given course, and this provides instructors with a
multitude of choices in tailoring their course to meet their own
goals and the needs of their students. The answers to almost all
of these problems can be found in the pages at the back of the
book; full solutions are in either the Student’s Solution Manual
or the Instructor’s Solution Manual.
While we make numerous references to the use of
technology, we do so without limiting instructor freedom to
use as much, or as little, technology as they desire. Appropriate
technologies include advanced graphing calculators (TI
Nspire), a spreadsheet (Excel), web-based resources (applets),
computer algebra systems, (Maple, Mathematica, Sage),
scientific computation systems (MATLAB), or traditional
programming (FORTRAN, Javascript, Python). Problems
marked with a G are ones we believe are best approached with
a graphical tool; those marked with a N are best solved with the
use of a numerical tool. Instructors should consider setting their
own policies, consistent with their interests and intents about
student use of technology when completing assigned problems.
Many problems in this book are best solved through
a combination of analytic, graphic, and numeric methods.
Pencil-and-paper methods are used to develop a model that
is best solved (or analyzed) using a symbolic or graphic
tool. The quantitative results and graphs, frequently produced
using computer-based resources, serve to illustrate and to
clarify conclusions that might not be readily apparent from
a complicated explicit solution formula. Conversely, the
vii
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October 12, 2016
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PREFACE
implementation of an efficient numerical method to obtain
an approximate solution typically requires a good deal of
preliminary analysis–to determine qualitative features of the
solution as a guide to computation, to investigate limiting
or special cases, or to discover ranges of the variables or
parameters that require an appropriate combination of both
analytic and numeric computation. Good judgment may well
be required to determine the best choice of solution methods
in each particular case. Within this context we point out that
problems that request a “sketch” are generally intended to
be completed without the use of any technology (except your
writing device).
We believe that it is important for students to understand
that (except perhaps in courses on differential equations) the
goal of solving a differential equation is seldom simply to
obtain the solution. Rather, we seek the solution in order to
obtain insight into the behavior of the process that the equation
purports to model. In other words, the solution is not an end
in itself. Thus, we have included in the text a great many
problems, as well as some examples, that call for conclusions
to be drawn about the solution. Sometimes this takes the form
of finding the value of the independent variable at which the
solution has a certain property, or determining the long-term
behavior of the solution. Other problems ask for the effect of
variations in a parameter, or for the determination of all values
of a parameter at which the solution experiences a substantial
change. Such problems are typical of those that arise in the
applications of differential equations, and, depending on the
goals of the course, an instructor has the option of assigning as
few or as many of these problems as desired.
Readers familiar with the preceding edition will observe
that the general structure of the book is unchanged. The
minor revisions that we have made in this edition are in
many cases the result of suggestions from users of earlier
editions. The goals are to improve the clarity and readability of
our presentation of basic material about differential equations
and their applications. More specifically, the most important
revisions include the following:
1. Chapter 1 has been rewritten. Instead of a separate section
on the History of Differential Equations, this material
appears in three installments in the remaining three
section.
2. Additional words of explanation and/or more explicit
details in the steps in a derivation have been added
throughout each chapter. These are too numerous and
widespread to mention individually, but collectively they
should help to make the book more readable for many
students.
3. There are about forty new or revised problems scattered
throughout the book. The total number of problems has
been reduced by about 400 problems, which are still
available through WileyPlus, leaving about 1600 problems
in print.
4. There are new examples in Sections 2.1, 3.8, and 7.5.
5. The majority (is this correct?) of the figures have been
redrawn, mainly by the use full color to allow for easier
identification of critical properties of the solution. In
addition, numerous captions have been expanded to clarify
the purpose of the figure without requiring a search of the
surrounding text.
6. There are several new references, and some others have
been updated.
The authors have found differential equations to be a
never-ending source of interesting, and sometimes surprising,
results and phenomena. We hope that users of this book, both
students and instructors, will share our enthusiasm for the
subject.
William E. Boyce and Douglas B. Meade
Watervliet, New York and Columbia, SC
29 August 2016
Supplemental Resources for
Instructors and Students
An Instructor’s Solutions Manual, ISBN 978-1-119-16976-5,
includes solutions for all problems not contained in the Student
Solutions Manual.
A Student Solutions Manual, ISBN 978-1-119-16975-8,
includes solutions for selected problems in the text.
A Book Companion Site, www.wiley.com/college/boyce,
provides a wealth of resources for students and instructors,
including
• PowerPoint slides of important definitions, examples, and
•
•
•
theorems from the book, as well as graphics for presentation
in lectures or for study and note taking.
Chapter Review Sheets, which enable students to test their
knowledge of key concepts. For further review, diagnostic
feedback is provided that refers to pertinent sections in the
text.
Mathematica, Maple, and MATLAB data files for selected
problems in the text providing opportunities for further
exploration of important concepts.
Projects that deal with extended problems normally not
included among traditional topics in differential equations,
many involving applications from a variety of disciplines.
These vary in length and complexity, and they can be
assigned as individual homework or as group assignments.
A series of supplemental guidebooks, also published by John
Wiley & Sons, can be used with Boyce/DiPrima/Meade in
order to incorporate computing technologies into the course.
These books emphasize numerical methods and graphical
analysis, showing how these methods enable us to interpret
solutions of ordinary differential equations (ODEs) in the real
world. Separate guidebooks cover each of the three major
mathematical software formats, but the ODE subject matter is
the same in each.
• Hunt, Lipsman, Osborn, and Rosenberg, Differential
Equations with MATLAB , 3rd ed., 2012, ISBN 978-1-11837680-5
Boyce 9131 FM 2
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PREFACE
• Hunt, Lardy, Lipsman, Osborn, and Rosenberg, Differential
•
Equations with Maple, 3rd ed., 2008, ISBN 978-0-47177317-7
Hunt, Outing, Lipsman, Osborn, and Rosenberg,
Differential Equations with Mathematica, 3rd ed., 2009,
ISBN 978-0-471-77316-0
WileyPLUS, is loaded with all of the supplements above,
and it also features
• The E-book, which is an exact version of the print text
•
WileyPLUS
WileyPLUS is an innovative, research-based
online environment for effective teaching and learning.
WileyPLUS builds students’ confidence because it takes
the guesswork out of studying by providing students with a
clear roadmap: what to do, how to do it, if they did it right.
Students will take more initiative so you’ll have greater impact
on their achievement in the classroom and beyond.
ix
•
•
but also features hyperlinks to questions, definitions, and
supplements for quicker and easier support.
Guided Online (GO) Exercises, which prompt students to
build solutions step-by-step. Rather than simply grading
an exercise answer as wrong, GO problems show students
precisely where they are making a mistake.
Homework management tools, which enable instructors
easily to assign and grade questions, as well as to gauge
student comprehension.
QuickStart pre-designed reading and homework assign
ments. Use them as is, or customize them to fit the needs of
your classroom.
Acknowledgments
It is a pleasure to express my appreciation to the many people who
have generously assisted in various ways in the preparation of this
book.
To the individuals listed below, who reviewed the manuscript
and/or provided valuable suggestions for its improvement:
Irina Gheorghiciuc, Carnegie Mellon University
Bernard Brooks, Rochester Institute of Technology
James Moseley, West Virginia University
D. Glenn Lasseigne, Old Dominion University
Stephen Summers, University of Florida
Fabio Milner, Arizona State University
Mohamed Boudjelkha, Rensselaer Polytechnic Institute
Yuval Flicker, The Ohio State University
Y. Charles Li, University of Missouri, Columbia
Will Murray, California State University, Long Beach
Yue Zhao, University of Central Florida
Vladimir Shtelen, Rutgers University
Zhilan Feng, Purdue University
Mathew Johnson, University of Kansas
Bulent Tosun, University of Alabama
Juha Pohjanpelto, Oregon State University
Patricia Diute, Rochester Institute of Technology
Ning Ju, Oklahoma State University
Ian Christie, West Virginia University
Jonathan Rosenberg, University of Maryland
Irina Kogan, North Carolina State University
To our colleagues and students at Rensselaer and The University
of South Carolina, whose suggestions and reactions through the years
have done much to sharpen our knowledge of differential equations, as
well as our ideas on how to present the subject.
To those readers of the preceding edition who called errors or
omissions to our attention.
To Tom Polaski (Winthrop University), who is primarily
responsible for the revision of the Instructor’s Solutions Manual and
the Student Solutions Manual.
To Mark McKibben (West Chester University), who checked the
answers in the back of the text and the Instructor’s Solutions Manual
for accuracy, and carefully checked the entire manuscript.
To the editorial and production staff of John Wiley & Sons, who
have always been ready to offer assistance and have displayed the
highest standards of professionalism.
The last, but most important, people we want to thank are our
wives: Elsa, for discussing questions both mathematical and stylistic
and above all for her unfailing support and encouragement, and Betsy,
for her encouragement, patience and understanding.
WILLIAM E. BOYCE AND DOUGLAS B. MEADE
Brief Contents
PREFACE
vii
1
Introduction 1
2
First-Order Differential Equations 24
3
Second-Order Linear Differential Equations 103
4
Higher-Order Linear Differential Equations 169
5
Series Solutions of Second-Order Linear Equations 189
6
The Laplace Transform 241
7
Systems of First-Order Linear Equations 281
8
Numerical Methods 354
9
Nonlinear Differential Equations and Stability 388
10
Partial Differential Equations and Fourier Series 463
11
Boundary Value Problems and Sturm-Liouville Theory 529
ANSWERS TO PROBLEMS
INDEX
x
60Ǚ
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Contents
PREFACE
vii
1 Introduction
1.1
1.2
1.3
4.3
4.4
1
Some Basic Mathematical Models; Direction
Fields 1
Solutions of Some Differential Equations 9
Classification of Differential Equations 16
2 First-Order Differential
Equations
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
24
Linear Differential Equations; Method of
Integrating Factors 24
Separable Differential Equations 33
Modeling with First-Order Differential
Equations 39
Differences Between Linear and Nonlinear
Differential Equations 51
Autonomous Differential Equations and
Population Dynamics 58
Exact Differential Equations and Integrating
Factors 70
Numerical Approximations: Euler’s Method 76
The Existence and Uniqueness Theorem 83
First-Order Difference Equations 91
3 Second-Order Linear Differential
Equations
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
103
Homogeneous Differential Equations with
Constant Coefficients 103
Solutions of Linear Homogeneous Equa…
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