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Differential Equations 4th Edition Student Format. We then move directly from first-order equations to systems of first-order differential equations. Rather than consider second-order equations separately, we convert these equations to first-order systems.
When these equations are viewed as systems, we are able to use qualitative and numerical techniques more readily.
Of course, we then use the information about these systems gleaned from these techniques to recover information about the solutions of the original equation. We also begin the treatment of systems with a general approach.
We do not immediately restrict our attention to linear systems. Qualitative and numerical techniques work just as easily on nonlinear systems, and one can proceed a long way toward understanding solutions without resorting to algebraic techniques. However, qualitative ideas do not tell the whole story, and we are led naturally to the idea of linearization.
With this background in the fundamental geometric and qualitative concepts, we then discuss linear systems in detail. Not only do we emphasize the formula for the general solution of a linear system, but also the geometry of its solution curves and its relationship to the eigenvalues and eigenvectors of the system.
While our study of systems requires the minimal use of some linear algebra, it is definitely not a prerequisite. Because we deal primarily with two-dimensional systems, Copyright Cengage Learning. In the process, we give considerable insight into the geometry of eigenvectors and eigenvalues. These topics form the core of our approach. However, there are many additional topics that one would like to cover in the course. Consequently, we have included discussions of forced second-order equations, nonlinear systems, Laplace transforms, numerical methods, and discrete dynamical systems.
In Appendix A, we even have a short discussion of Riccati and Bernoulli equations, and Appendix B is an ultra-lite treatment of power series methods. In Appendix B we take the point of view that power series are an algebraic way of finding approximate solutions much like numerical methods.
Occasional surprises, such as Hermite and Legendre polynomials, are icing on the cake. Although some of these topics are quite traditional, we always present them in a manner that is consistent with the philosophy developed in the first half of the text. Good labs are tough to write and to grade, but we feel that the benefit to students is extraordinary.
Changes in the Fourth Edition This revision has been our most extensive since we published the first edition in In Chapter 1, the table of contents remains the same.
However, many new exercises have been added, and they often introduce models that are new to the text. For example, the theta model for the spiking of a neuron appears in the exercise sets of Sections 1. The concept of a time constant is introduced in Section 1. The velocity of a freefalling skydiver is discussed in three exercise sets.
In Section 1. Chapter 2 has undergone a complete overhaul. We added a section Section 2. We include this topic for two reasons. First, many of our students had first-hand experience with the H1N1 pandemic in — Second, many users of the preliminary edition liked the fact that we discussed nullclines in Chapter 2. Section 2. Chapter 2 now has eight sections rather than five. Sections 2. This model is so important that it deserves a section of its own rather than being buried at the end of a section as it was in previous editions.
The remaining analytic techniques that we presented in the previous editions can now be found in Section 2. The Existence and Uniqueness Theorem for systems along with its consequences has its own section Section 2. This material is presented in smaller sections to give the instructor more flexibility to pick and choose topics from Chapter 2.
Only Sections 2. Chapter 2 has always been the most difficult one to teach, and now instructors can cover as many or as few sections from Chapter 2 as they see fit. Pathways Through This Book There are a number of possible tracks that instructors can follow in using this book. Chapters 1—3 form the core with the possible exception of Sections 2. Most of the later chapters assume familiarity with this material.
Certain sections such as Section 1. However, the material on phase lines and phase planes, qualitative analysis, and solutions of linear systems is central. A typical track for an engineering-oriented course would follow Chapters 1—3 perhaps skipping Sections 1. Appendix A changing variables can be covered at the end of Chapter 1. These chapters will take roughly twothirds of a semester.
The final third of the course might cover Sections 4. Chapters 4 and 5 are independent of each other and can be covered in either order. In particular, Section 5.
Appendix B power series goes well after Chapter 4. Incidentally, it is possible to cover Sections 6. As we have learned from our colleagues in the College of Engineering at Boston University, some engineering programs teach a circuit theory course that uses the Laplace transform early in the course.
Consequently, Sections 6. However, if possible, we recommend waiting to cover Chapter 6 entirely until after the material in Sections 4. Instructors can substitute material on discrete dynamics Chapter 8 for Laplace transforms. A course for students with a strong background in physics might involve more of Chapter 5, including a treatment of systems that are Hamiltonian Section 5. A course geared toward applied mathematics might include a more detailed discussion of numerical methods Chapter 7.
We have sample syllabi contributed by users at various institutions as well as information about workshops and seminars dealing with the teaching of differential equations. We also maintain a list of errata. Solution Builder, available Copyright Cengage Learning. PREFACE xi to instructors who have adopted the text for class use, creates customized, secure PDF copies of solutions matched exactly to the the exercises assigned for class.
The goal of that project was to rethink the traditional, sophomore-level differential equations course. We are especially thankful for that support. Paul Blanchard Robert L. For this edition, we owe particular thanks to one individual, Brian Persaud. Over the course of almost two years, he was there when we needed him.
He basically did all of the production work that was done at Boston University. He researched countless topics and gave us valuable feedback at every step in the process.
Thanks, Brian. Doug Wright on the second edition, Sebastian Marotta and Dan Look on the third edition, Brian has left his mark on the text in many positive ways.
We also had help with the graphics programming from Nick Benes.
Nick is currently serving in Afghanistan, but we have the light on in anticipation of his safe return. When we needed some last-minute accuracy checking, Dan Cuzzocreo was the man. We also thank Mark Kramer for giving us copies of his notes on the theta model for the spiking of a neuron.
He saved us countless hours during the formating of the text and the production of the figures. Much of the production work, solutions to exercises, accuracy checking and rendering of pictures was done by former Boston University graduate students Bill Basener, Lee DeVille, and Stephanie Jones during the production of the first edition.
They spent many long days and nights in an alternately too-hot-or-too-cold windowless computer lab to bring this book to completion. Many other individuals at Boston University have made important contributions. In particular, our teaching assistants Duff Campbell, Michael Hayes, Eileen Lee, and Clara Bodelon had to put up with the headaches associated with our experimentation. Duff also advised us during the development of our power series appendix for the third edition, and he continues to provide valuable feedback whenever he teaches differential equations using this book.
We received support from many of our colleagues at Boston University and at other institutions. It was a special pleasure for us to work closely with colleagues in the College of Engineering—Michael Ruane who often coordinates the circuits course , Moe Wasserman who permitted one of the authors to audit his course , and John Baillieul a member of our advisory board. All contributed their scarce time during workshops and trips to Boston University.
This software truly enhances the teaching and learning of differential equations. As you will see while using these programs, Hu has the ability to visualize differential equations in a way that only an artist as he is can. It is always a pleasure to work with him.
We are also grateful to John Cantwell, Jean Marie McDill, Steven Strogatz, and Beverly West for their many helpful discussions about incorporating technology in the differential equations course. Their pathbreaking work with in-class and hands-on software demonstrations has been an inspiration, both to us and to our students.
It has had a major impact on the way we teach and think about this subject.
In particular we thank Lee Stayton for managing the conversion and Dusan A. Koljensic for designing the new interface to the tools. We were pleased that so many of our colleagues outside of Boston University were willing to help us with this project. Bill Krohn gave us valuable advice regarding our exposition of Laplace transforms, and Bruce Elenbogen did a thorough reading of early drafts of the beginning chapters.
We have been pleased with the reception given to this text. We particularly wish to thank the many people who have generously shared their ideas and suggestions. Also, thanks to all who caught the typo we made in the third edition. It is corrected in this edition. Thoughtful and insightful reviews have also been a tremendous help in the evolution of this text from preliminary to fourth edition, and we thank all those who took the time to give us valuable feedback.
From October through December, he traveled extensively, and he was the guest of many organizations and individuals.