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# Differential Equations with Boundary-Value Problems, 8th Edition

### Overview

DIFFERENTIAL EQUATIONS WITH BOUNDARY-VALUE PROBLEMS, 8th Edition strikes a balance between the analytical, qualitative, and quantitative approaches to the study of differential equations. This proven and accessible book speaks to beginning engineering and math students through a wealth of pedagogical aids, including an abundance of examples, explanations, "Remarks" boxes, definitions, and group projects. Written in a straightforward, readable, and helpful style, the book provides a thorough treatment of boundary-value problems and partial differential equations.

### Features and Benefits

• The four-color design adds depth of meaning to all of the graphics, particularly three-dimensional pieces and visuals that involve multiple curves in a graph. The author directed the creation of each piece of art to ensure that it is as mathematically correct as the text.
• Extended homework problems at the end of selected section exercises were submitted and classroom-tested by members of the teaching math community.
• The development of material in this text progresses intuitively, and explanations are clear and concise. Exercises reinforce and build on chapter content.
• This text guides students through material necessary to progress to the next level of study; its clear presentation and mathematical precision make it an excellent reference tool in future courses.
• While this text is time-tested and widely accepted, it has remained current with the addition of new exercises, such features as "Contributed Problems," and the enhanced four-color presentation.

1. INTRODUCTION TO DIFFERENTIAL EQUATIONS.
Definitions and Terminology. Initial-Value Problems. Differential Equations as Mathematical Models. Chapter 1 in Review.
2. FIRST-ORDER DIFFERENTIAL EQUATIONS.
Solution Curves Without a Solution. Separable Variables. Linear Equations. Exact Equations and Integrating Factors. Solutions by Substitutions. A Numerical Method. Chapter 2 in Review.
3. MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS.
Linear Models. Nonlinear Models. Modeling with Systems of First-Order Differential Equations. Chapter 3 in Review.
4. HIGHER-ORDER DIFFERENTIAL EQUATIONS.
Preliminary Theory-Linear Equations. Reduction of Order. Homogeneous Linear Equations with Constant Coefficients. Undetermined Coefficients-Superposition Approach. Undetermined Coefficients-Annihilator Approach. Variation of Parameters. Cauchy-Euler Equation. Solving Systems of Linear Differential Equations by Elimination. Nonlinear Differential Equations. Chapter 4 in Review.
5. MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS.
Linear Models: Initial-Value Problems. Linear Models: Boundary-Value Problems. Nonlinear Models. Chapter 5 in Review.
6. SERIES SOLUTIONS OF LINEAR EQUATIONS.
Review of Power Series Solutions About Ordinary Points. Solutions About Singular Points. Special Functions. Chapter 6 in Review.
7. LAPLACE TRANSFORM.
Definition of the Laplace Transform. Inverse Transform and Transforms of Derivatives. Operational Properties I. Operational Properties II. Dirac Delta Function. Systems of Linear Differential Equations. Chapter 7 in Review.
8. SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS.
Preliminary Theory. Homogeneous Linear Systems. Nonhomogeneous Linear Systems. Matrix Exponential. Chapter 8 in Review.
9. NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS.
Euler Methods. Runge-Kutta Methods. Multistep Methods. Higher-Order Equations and Systems. Second-Order Boundary-Value Problems. Chapter 9 in Review.
10. PLANE AUTONOMOUS SYSTEMS.
Autonomous Systems. Stability of Linear Systems. Linearization and Local Stability. Autonomous Systems as Mathematical Models. Chapter 10 in Review.
11. ORTHOGONAL FUNCTIONS AND FOURIER SERIES.
Orthogonal Functions. Fourier Series and Orthogonal Functions. Fourier Cosine and Sine Series. Sturm-Liouville Problem. Bessel and Legendre Series. Chapter 11 in Review.
12. BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES.
Separable Partial Differential Equations. Classical PDE's and Boundary-Value Problems. Heat Equation. Wave Equation. Laplace's Equation. Nonhomogeneous Boundary-Value Problems. Orthogonal Series Expansions. Higher-Dimensional Problems. Chapter 12 in Review.
13. BOUNDARY-VALUE PROBLEMS IN OTHER COORDINATE SYSTEMS.
Polar Coordinates. Polar and Cylindrical Coordinates. Spherical Coordinates. Chapter 13 in Review.
14. INTEGRAL TRANSFORM METHOD.
Error Function. Laplace Transform. Fourier Integral. Fourier Transforms. Chapter 14 in Review.
15. NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS.
Laplace's Equation. Heat Equation. Wave Equation. Chapter 15 in Review.
Appendix I: Gamma Function.
Appendix II: Matrices.
Appendix III: Laplace Transforms.

### What's New

• Exercises in this edition have been updated to better test and challenge students. Revisions are based on reviewer and user input, as well as the author's understanding of the course goals.
• Chapter 6 now starts with a new section on the review of power series.
• This edition contains an optional section on Green's functions in Chapter 4 and for those who have extra time in their course to consider this elegant application of variation of parameters in the solution of initial-value and boundary-value problems.
• New projects have been added to the text to show how the concepts of Differential Equations will apply to students' intended careers.
• Exclusively from Cengage Learning, Enhanced WebAssign® now offers an extensive online program for Zill/Wright, Differential Equations with Boundary-Value Problems, 8th Edition. EWA is the most widely used homework system in higher education. Available for this title, EWA allows you to assign, collect, grade, and record assignments via the web. This proven homework system includes links to textbook sections, video examples, and problem specific tutorials. EWA is more than a homework system - it is a complete learning system for students.

### Bundle: Text + Enhanced WebAssign with eBook LOE Printed Access Card for One-Term Math and Science + Custom Enrichment Module: Enhanced WebAssign - Start Smart Guide for Students

ISBN-10: 1133802796  | ISBN-13: 9781133802792

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### Bundle: Text + Student Solutions Manual

ISBN-10:  1133804047  | ISBN-13:  9781133804048

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ISBN-10:  1133540805  | ISBN-13:  9781133540809

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ISBN-10:  1305434056  | ISBN-13:  9781305434059

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ISBN-10:  1305588932  | ISBN-13:  9781305588936

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### ePack: Printed Text + Enhanced WebAssign - Start Smart Guide for Students + Enhanced WebAssign Instant Access for Advanced Math, Single-Term Courses

ISBN-10:  1305593413 | ISBN-13:  9781305593411

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## Supplements

All supplements have been updated in coordination with the main title. Select the main title's "About" tab, then select "What's New" for updates specific to title's edition.

### Instructor Supplements

#### Solution Builder  (ISBN-10: 1133491936 | ISBN-13: 9781133491934)

This online instructor database offers complete worked solutions to all exercises in the text, allowing you to create customized, secure solutions printouts (in PDF format) matched exactly to the problems you assign in class. Visit Cengage.com/solutionbuilder for more information.

#### ExamView®  (ISBN-10: 1133492290 | ISBN-13: 9781133492290)

Available on CD, ExamView® testing software allows instructors to quickly create, deliver, and customize tests for class in print and online formats, and features automatic grading. This electronic test bank includes hundreds of questions customized directly to the text, with all questions also provided in PDF and Microsoft® Word formats for instructors who opt not to use the software component.

#### Student Solutions Manual  (ISBN-10: 1133491952 | ISBN-13: 9781133491958)

This manual contains fully worked-out solutions to all of the odd-numbered exercises in the text, giving students a way to check their answers and ensure that they took the correct steps to arrive at an answer.

List Price = \$116.95  | CengageBrain Price = \$116.95  | College Bookstore Wholesale Price = \$88.00

List Price = \$94.00  | CengageBrain Price = \$94.00  | College Bookstore Wholesale Price = \$94.00

#### Enhanced WebAssign - Start Smart Guide for Students  (ISBN-10: 0495384798 | ISBN-13: 9780495384793)

If your instructor has chosen to package Enhanced WebAssign with your text, this manual will help you get up and running quickly with the Enhanced WebAssign system so you can study smarter and improve your performance in class.

List Price = \$50.95  | CengageBrain Price = \$50.95  | College Bookstore Wholesale Price = \$38.25

### Student Supplements

#### Student Solutions Manual  (ISBN-10: 1133491952 | ISBN-13: 9781133491958)

Go beyond the answers--see what it takes to get there and improve your grade! This manual provides worked-out, step-by-step solutions to the odd-numbered problems in the text, giving you the information you need to truly understand how these problems are solved. Each section begins with a list of key terms and concepts. The solutions sections also include hints and examples to guide you to greater understanding.

List Price = \$116.95  | CengageBrain Price = \$116.95  | College Bookstore Wholesale Price = \$88.00

List Price = \$94.00  | CengageBrain Price = \$94.00  | College Bookstore Wholesale Price = \$94.00

#### Enhanced WebAssign - Start Smart Guide for Students  (ISBN-10: 0495384798 | ISBN-13: 9780495384793)

If your instructor has chosen to package Enhanced WebAssign with your text, this manual will help you get up and running quickly with the Enhanced WebAssign system so you can study smarter and improve your performance in class.

List Price = \$50.95  | CengageBrain Price = \$50.95  | College Bookstore Wholesale Price = \$38.25

## Meet the Author

### Dennis G. Zill

Dennis G. Zill is professor of mathematics at Loyola Marymount University. His interests are in applied mathematics, special functions, and integral transforms. Dr. Zill received his Ph.D. in applied mathematics and his M.S. from Iowa State University in 1967 and 1964, respectively. He received his B.A. from St. Mary's in Winona, Minnesota, in 1962. Dr. Zill also is former chair of the Mathematics Department at Loyola Marymount University. He is the author or co-author of 13 mathematics texts.

### Warren S. Wright

Warren S. Wright received his M.A. in mathematics from the University of Southern California and is currently professor of mathematics at Loyola Marymount University in Los Angeles. Wright has previously coauthored two textbooks with Dennis Zill and has authored numerous solutions manuals to accompany Zill's texts.