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**WebAssign** is a powerful digital solution designed by educators to help students learn maths and science, not just do homework. Instructors get the flexibility to define mastery thresholds and manage student learning aids down to the question level giving students the support they need to learn when they need it. **WebAssign** provides extensive content, instant assessment, and superior support.

DIFFERENTIAL EQUATIONS WITH BOUNDARY-VALUE PROBLEMS, 9E, INTERNATIONAL METRIC EDITION strikes a balance between the analytical, qualitative, and quantitative approaches to the study of Differential Equations. This proven text speaks to students of varied majors through a wealth of pedagogical aids, including an abundance of examples, explanations, "Remarks" boxes, and definitions. Written in a straightforward, readable, and helpful style, the book provides a thorough overview of the topics typically taught in a first course in Differential Equations as well as an introduction to boundary-value problems and partial Differential Equations.

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.

Answers for Selected Odd-Numbered Problems.

- Exercises in this edition have been updated to better test and challenge students.
- A new design helps students quickly identify resources in the text and emphasizes the application of Differential Equations in the world around them.

- 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 differential equations instructors.
- 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, and the enhanced four-color presentation.

**Dennis G. Zill**

Loyola Marymount University

Dennis Zill received a PhD in Applied Mathematics from Iowa State University, and is a former professor of Mathematics at Loyola Marymount University in Los Angeles, Loras College in Iowa, and California Polytechnic State University. He is also the former chair of the Mathematics department at Loyola Marymount University, where he currently holds a rank as Professor Emeritus of Mathematics. Zill holds interests in astronomy, modern literature, music, golf, and good wine, while his research interests include Special Functions, Differential Equations, Integral Transformations, and Complex Analysis.

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