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Publisher:

Brooks/Cole

Publication Date:

2009

Number of Pages:

526

Format:

Hardcover

Edition:

7

Price:

192.95

ISBN:

9780495108368

Category:

Textbook

[Reviewed by , on ]

Michael Berg

03/18/2009

Zill-Cullen, *Differential Equations with Boundary-Value Problems*, is a solid text in differential equations and methods of applied mathematics for advanced undergraduates, be they engineering majors, future physicists, or fledgling mathematicians. At my university, which was also home to the two authors for many years, Zill-Cullen has for many years been the book of choice in our courses as described (the second of which I always think of as PDE, really), with the two authors prominent in the rotation. However, I have also taught “Methods” (i.e. Methods of Applied Mathematics) half a dozen times over the years, and want to put in a plug for this fine book myself.

Zill, who retired from a long career as a professor only a year ago, has a phenomenal success record as a writer of calculus and differential equations books, with the book under review at one point (in the 1990s, I believe) climbing the charts to number two of its type nationwide. Thus, the present seventh edition is a highly evolved version of a winning textbook with a seriously successful track record. The book is compactly written, with fine, clear explanations of everything it deals with, from ODE (the usual fare, including such stalwarts as the method of undetermined coefficients and the method of variation of parameters, Cauchy-Euler equations, reduction of order, etc.) to separable PDE (with the big three — heat, wave, Laplace — taking central stage), and with Laplace (*bis*: it’s his transform), Runge-Kutta, Fourier, Bessel, Legendre, and Sturm-Liouville making appearances besides. In other words, Zill and Cullen cover quite a bit of material, more than one would cover in the usual sequence, and the book does it very effectively.

Being geared toward mixed audiences, the emphasis is taken off *Sätze und Beweisen* somewhat, even though rigor is properly championed: everything is there, but the fact that we’re dealing with engineers is never far from the authors’ minds. Indeed, over the years I have taken less time with theorems and proofs and progressively more with examples and illustrations, simply because of the abundance of students in front of me who wouldn’t know a Wronskian if it bit them in the fleshy parts. Oh well…

Nonetheless, I have found *Differential Equations with Boundary-Value Problems* perfectly suited to my teaching tasks, and the same is true for my colleagues, all of whose styles are considerably more contemporary than my own Jurassic approach. Indeed, the book under review comes equipped with a lot of material on modeling (first order as well as higher order DE), computer connections (e.g., “lab assignments” — which I avoid like the plague), and so on. The chapters, as well as individual sections, are introduced *via* brief passages about what’s coming up, and the problem sets are fine and ramified. And each chapter closes with very good review problems. It’s a well designed book.

On a more personal note, I never took a course on PDE as an undergraduate so when my turn came many years ago to teach this material for the first time, Zill-Cullen was my own text from which to learn all this material, trying hard to stay ahead of the students. It came to pass that at that time I was writing my first serious research paper, and it presently required some manoeuvreing with, yes, PDE. I found myself able to write out the most important theorem in that paper using what I had learnt only a short time before from *Differential Equations with Boundary-Value Problems*. Furthermore, my currently burgeoning work on theta functions has taken me back to the heat equation, particularly because of my much increased familiarity with it from my teaching “Methods” twice in the last three years, always from *Differential Equations with Boundary-Value Problems*. I owe this book a lot.

Lastly, I want to acknowledge Mike Cullen, God rest his soul, and Dennis Zill, now *emeritus*. When I was hired at my university Cullen was chair and Zill was head of the hiring committee. Cullen, who passed away from brain-cancer almost a decade ago, was an award-winning teacher, whose mastery of the indicated material was truly impressive; he was also one of the funniest people I have ever known (I could tell you stories…). And I have spent nigh on twenty years as a colleague to Dennis Zill, who for many years occupied an office across the hall from me. Early on, our relationship underwent several quantum jumps (in the right directions) due to the fact that I played music too loudly in my office. To wit: One morning Dennis appeared in my doorway looking very intent on saying something; I figured my blasting Isolde’s *Liebestod* through the building had raised his ire. *Au contraire!* He soon cracked a smile and said that if I was going to play Wagner I should turn the volume up a bit more so that he could hear it better. Ah, culture! I wish he hadn’t retired: I miss him.

It’s a fine book. Use it. You won’t be sorry.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

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.

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.

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