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Rules

1. Constant: d dx

c = 0 2. Constant Multiple: d dx

cf ( x ) = c f ( x )

. Sum: d dx

[ f ( x ) ± g ( x )] =

f ( x ) ± g ( x ) 4. Product: d dx

f ( x ) g ( x ) = f ( x ) g ( x ) + g ( x ) f ( x )

5. Quotient: d dx

f ( x ) g ( x )

g ( x ) f ( x ) f ( x ) g ( x ) [ g ( x )]^2

6. Chain: d dx

f ( g ( x )) = f ( g ( x )) g ( x )

7. Power: d dx

x n^ = nx n ^1 8. Power: d dx

[ g ( x )] n^ = n [ g ( x )] n^^1 g ( x )

Functions

Trigonometric:

9. d dx

sin x = cos x 10. d dx

cos x = sin x 11. d dx

tan x = sec 2 x

d dx

cot x = csc 2 x 13.

d dx

sec x = sec x tan x 14.

d dx

csc x = csc x cot x

Inverse trigonometric:

15.

d dx

sin^1 x =

1 x^2

d dx

cos^1 x =

1 x^2

d dx

tan^1 x =

1 + x^2

18. d dx

cot^1 x =

1 + x^2

d dx

sec^1 x =

x x^2 1

d dx

csc^1 x =

x x^2 1 Hyperbolic:

21. d dx

sinh x = cosh x 22. d dx

cosh x = sinh x 23. d dx

tanh x = sech 2 x

d dx

coth x = csch 2 x 25. d dx

sech x = sech x tanh x 26. d dx

csch x = csch x coth x

Inverse hyperbolic:

27.

d dx

sinh^1 x =

x^2 + 1

d dx

cosh^1 x =

x^2 1

d dx

tanh^1 x =

1 x^2

30. d dx

coth^1 x =

1 x^2

d dx

sech^1 x =

x 1 x^2

d dx

csch^1 x =

x x^2 + 1 Exponential:

33. d dx

e x^ = e x^ 34. d dx

b x^ = b x^ (ln b )

Logarithmic:

35. d dx

ln x =

x

d dx

log b x =

x (ln b )

REVIEW OF DIFFERENTIATION

TABLE OF LAPLACE TRANSFORMS

f ( t )

1. 1 2. t 3. t n^ n a positive integer 4. t
1/ 5. t 1/ 6. t a 7. sin kt 8. cos kt 9. sin^2 kt 10. cos^2 kt 11. e at 12. sinh kt 13. cosh kt 14. sinh^2 kt 15. cosh^2 kt 16. te at 17. t n^ e at^ n a positive integer 18. e at^ sin kt 19. e at^ cos kt s
a ( s
a )^2  k^2

k ( s
a )^2  k^2

n! ( s
a ) n ^1 ,

1 ( s
a )^2

s^2
2 k^2 s ( s^2
4 k^2 )

2 k^2 s ( s^2
4 k^2 )

s s^2
k^2

k s^2
k^2

1 s
a

s^2  2 k^2 s ( s^2  4 k^2 )

2 k^2 s ( s^2  4 k^2 )

s s^2  k^2

k s^2  k^2

(   1) s^ ^ ^1 , a 
1

1  2 s 3/

B

 s

n! sn ^1 ,

1 s^2

1 s

{ f ( t )}  F ( s ) f ( t )

20. e at^ sinh kt 21. e at^ cosh kt 22. t sin kt 23. t cos kt 24. sin kt  kt cos kt 25. sin kt
kt cos kt 26. t sinh kt 27. t cosh kt

28.

29.

30. 1
cos kt 31. kt
sin kt

32.

33.

34. sin kt sinh kt 35. sin kt cosh kt 36. cos kt sinh kt 37. cos kt cosh kt 38. J 0 ( kt ) 1 (^1) s^2  k^2

s^3 s^4  4 k^4

k ( s^2
2 k^2 ) s^4  4 k^4

k ( s^2  2 k^2 ) s^4  4 k^4

2 k^2 s s^4  4 k^4

s ( s^2  a^2 )( s^2  b^2 )

cos bt
cos at a^2
b^2

1 ( s^2  a^2 )( s^2  b^2 )

a sin bt
b sin at ab ( a^2
b^2 )

k^3 s^2 ( s^2  k^2 )

k^2 s ( s^2  k^2 )

s ( s
a )( s
b )

aeat^
bebt a
b

1 ( s
a )( s
b )

eat^
ebt a
b

s^2  k^2 ( s^2
k^2 )^2

2 ks ( s^2
k^2 )^2

2 k^3 ( s^2  k^2 )^2

2 ks^2 ( s^2  k^2 )^2

s^2
k^2 ( s^2  k^2 )^2

2 ks ( s^2  k^2 )^2

s
a ( s
a )^2
k^2

k ( s
a )^2
k^2

{ f ( t )}  F ( s )

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SEVENTH EDITION

DIFFERENTIAL

EQUATIONS

with Boundary-Value Problems

DENNIS G. ZILL

Loyola Marymount University

MICHAEL R. CULLEN

Late of Loyola Marymount University

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

v

5

4

• 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS CONTENTS - 1.1 Definitions and Terminology Preface xi - 1.2 Initial-Value Problems - 1.3 Differential Equations as Mathematical Models - CHAPTER 1 IN REVIEW
• 2 FIRST-ORDER DIFFERENTIAL EQUATIONS - 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 - 2.4 Exact Equations - 2.5 Solutions by Substitutions - 2.6 A Numerical Method - CHAPTER 2 IN REVIEW - MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS - 3.1 Linear Models - 3.2 Nonlinear Models - 3.3 Modeling with Systems of First-Order DEs - CHAPTER 3 IN REVIEW
• HIGHER-ORDER DIFFERENTIAL EQUATIONS vi ● CONTENTS
  • 4.1 Preliminary Theory—Linear Equations
    • 4.1.1 Initial-Value and Boundary-Value Problems
    • 4.1.2 Homogeneous Equations
    • 4.1.3 Nonhomogeneous Equations
  • 4.2 Reduction of Order
  • 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
  • 4.7 Cauchy-Euler Equation
  • 4.8 Solving Systems of Linear DEs by Elimination
  • 4.9 Nonlinear Differential Equations
  • CHAPTER 4 IN REVIEW
• 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
  • 5.2 Linear Models: Boundary-Value Problems
  • 5.3 Nonlinear Models
  • CHAPTER 5 IN REVIEW
• SERIES SOLUTIONS OF LINEAR EQUATIONS
  • 6.1 Solutions About Ordinary Points
    • 6.1.1 Review of Power Series
    • 6.1.2 Power Series Solutions
  • 6.2 Solutions About Singular Points
  • 6.3 Special Functions
    • 6.3.1 Bessel’s Equation
    • 6.3.2 Legendre’s Equation
  • CHAPTER 6 IN REVIEW
• 7 THE LAPLACE TRANSFORM CONTENTS ● vii
  • 7.1 Definition of the Laplace Transform
  • 7.2 Inverse Transforms and Transforms of Derivatives
    • 7.2.1 Inverse Transforms
    • 7.2.2 Transforms of Derivatives
  • 7.3 Operational Properties I
    • 7.3.1 Translation on the s -Axis
    • 7.3.2 Translation on the t -Axis
  • 7.4 Operational Properties II
    • 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
  • 7.6 Systems of Linear Differential Equations
  • CHAPTER 7 IN REVIEW
• 8 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
    • 8.2.3 Complex Eigenvalues
  • 8.3 Nonhomogeneous Linear Systems
    • 8.3.1 Undetermined Coefficients
    • 8.3.2 Variation of Parameters
  • 8.4 Matrix Exponential
  • CHAPTER 8 IN REVIEW
• 9 NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS
  • 9.1 Euler Methods and Error Analysis
  • 9.2 Runge-Kutta Methods
  • 9.3 Multistep Methods
  • 9.4 Higher-Order Equations and Systems
  • 9.5 Second-Order Boundary-Value Problems
  • CHAPTER 9 IN REVIEW
  • 10 PLANE AUTONOMOUS SYSTEMS viii ● CONTENTS
    • 10.1 Autonomous Systems
    • 10.2 Stability of Linear Systems
    • 10.3 Linearization and Local Stability
    • 10.4 Autonomous Systems as Mathematical Models
    • CHAPTER 10 IN REVIEW
  • 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES
    • 11.1 Orthogonal Functions
    • 11.2 Fourier Series
    • 11.3 Fourier Cosine and Sine Series
    • 11.4 Sturm-Liouville Problem
    • 11.5 Bessel and Legendre Series
      • 11.5.1 Fourier-Bessel Series
      • 11.5.2 Fourier-Legendre Series
    • CHAPTER 11 IN REVIEW
• 12 BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES - 12.1 Separable Partial Differential Equations - 12.2 Classical PDEs and Boundary-Value Problems - 12.3 Heat Equation - 12.4 Wave Equation - 12.5 Laplace’s Equation - 12.6 Nonhomogeneous Boundary-Value Problems - 12.7 Orthogonal Series Expansions - 12.8 Higher-Dimensional Problems - CHAPTER 12 IN REVIEW
• 13 BOUNDARY-VALUE PROBLEMS IN OTHER COORDINATE SYSTEMS CONTENTS ● ix - 13.1 Polar Coordinates - 13.2 Polar and Cylindrical Coordinates - 13.3 Spherical Coordinates - CHAPTER 13 IN REVIEW
• 14 INTEGRAL TRANSFORMS - 14.1 Error Function - 14.2 Laplace Transform - 14.3 Fourier Integral - 14.4 Fourier Transforms - CHAPTER 14 IN REVIEW
• 15 NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS - 15.1 Laplace’s Equation - 15.2 Heat Equation - 15.3 Wave Equation - CHAPTER 15 IN REVIEW
  • I Gamma Function APP- APPENDICES
  • II Matrices APP-
  • III Laplace Transforms APP-
  • Answers for Selected Odd-Numbered Problems ANS-
  • Index I-

xi

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 cho- sen 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 work book. By this I mean that mathemat- ics 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 ex- amples 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 solu- tion; 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 fol- lowed by the slow assimilation of understanding simply cannot be achieved without a struggle. Specifically for you, a Student Resource and Solutions Manual ( SRSM ) is avail- able as an optional supplement. In addition to containing solutions of selected prob- lems from the exercises sets, the SRSM has hints for solving problems, extra exam- ples, 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: [email protected]

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.

PREFACE

In case you are examining this text for the first time, Differential Equations with Boundary-Value Problems, 7th Edition, can be used for either a one-semester course in ordinary differential equations, or a two-semester course covering ordinary and partial differential equations. The shorter version of the text, A First Course in Differential Equations with Modeling Applications, 9th Edition , ends with Chapter 9. For a one-semester course, I assume that the students have successfully completed at least two-semesters of calculus. Since you are reading this, undoubt- edly 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 bal- ance between the analytical, qualitative, and quantitative approaches to the study of differential equations. As far as my “underlying philosophy” it is this: An under- graduate 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 con- tributed problems. These problems were class tested and submitted by in- structors 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 prob- lems 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 mathemati- cally 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 sys- tem. 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

7.6.5 Fifth figure in the section I feel that this system provides a clearer indication to where things are, with- out 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 (ac- companies Differential Equations with Boundary-Value Problems, 7e)) pro- vides reviews of important material from algebra and calculus, the solution of every third problem in each exercise set (with the exception of the Discussion 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.

bb

xii ●^ PREFACE

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 Tom Roe, South Dakota State University Kimmo I. Rosenthal, Union College Barbara Shabell, California Polytechnic State University

xiv ●^ PREFACE

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

PREFACE ●^ xv