Elementary Differential Equations with Boundary Value Problems
William Trench, Trinity University
Pub Date: 2013
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The text adequately covers the topics expected in an introduction to differential equations textbook. It seems to be quite comparable to other intro read more
The text adequately covers the topics expected in an introduction to differential equations textbook. It seems to be quite comparable to other intro to ODE books that I've seen (ones that students pay a lot of money to use). A couple of differences between this book and others are the use of variation of parameters, which is introduced early and used throughout the book, and the separation of application problems into their own chapters. Separating the application problems allows for instructors to skip these chapters if they wish. My personal preference is to integrate them into the other chapters, but separating them is a valid choice. A comprehensive index is included, and I particularly like that page numbers in the index are links to those pages in the text.
I have not, yet, used the text to teach a semester-long course, so don't feel prepared to answer this at this point. However, I looked closely at a couple of the chapters and found the content to be accurate.
The content of this course has been the same for many, many years, and I don't see it changing in the near future. That being said, some nice application problems are included in the text that are relevant to today's students. On another note, I know that some instructors feel strongly that technology should be included in a differential equations course, while others feel just as strongly that it should not. The author does a nice job of providing an adequate number of problems that don't require students' use of technology, while providing several others that do. These are marked clearly in the text so that the instructor can know at a glance.
The author claims that the text was written so that students can easily read it and states that he erred on the side of caution when deciding how much detail to include (in other words, the author claims that lots of details are provided, making it an easy text for students to read). I agree that it isn't difficult to read, but I would actually have liked to see it written at an even more elementary level. For example, in chapter 2, the author uses language like "y=0 is obviously a solution to the homogeneous equation" (page 30). From my own teaching experience, I can firmly say that this is not obvious to students at the beginning of a course. There are similar statements throughout as well as statements about things they "know" from their calculus classes. I don't think this is a fatal flaw in the book; proper instruction during class time can address these common student questions. However, I would have preferred to see more details provided in the first few chapters.
Since I have not read the entire book, I don't feel qualified to answer this definitively, but the chapters that I read carefully and the chapters that I've skimmed seem to be consistent.
There are 13 chapters, broken into smaller subsections. They seem to be appropriately named and are standard for intro to ODE books.
The text is organized in a clear fashion. The preface to the book nicely clarifies which chapters can be rearranged. For example, the book is written so that Fourier Solutions and Boundary Value Problems (Chapters 11, 12, and 13) can be covered in any order, as long as Chapter 5 (Linear Second Order Equations) is covered first.
The book has no interface issues that I noticed. Navigation was fairly easy, with some links to exercises as well as links to information on Wikipedia. My only trouble was when I clicked on one of these links, it wasn't always easy to go back to where I had been in the text. I later learned how to view the table of contents on the left side of my screen at all times, so this made this much easier. As mentioned earlier, I particularly like that the page numbers in the index are all links to those pages.
I did not find any grammatical errors in the text.
I found no issues with regards to cultural relevance.
The text gives a very thorough treatment of the topics in a traditional beginning course in ODE. read more
The text gives a very thorough treatment of the topics in a traditional beginning course in ODE.
The book is carefully written.
The topics are completely in line with the topics in the traditional course such as our Engineering Math IV Differential Equations. I don't envision changes in the basic material any time soon.
The book is very well written. The most difficult thing for an instructor will be in selection the portions of the text to include in a course. There is more there than can be carefully treated in one course.
The book is carefully written in the standard mathematical style.
The book was not written as electronic materials. While the .pdf production is beautiful and does have numerous hyperlinks, it is one long scroll...
The book is very well organized mathematically.
As a .pdf of a print book, the eBook is beautiful. But it is not modular and there is no "back" button for links. This is a general weakness of this technology.
It's fine. The questions here should have been: How's the math? and How are the applications? At a student level, the mathematical presentation is pretty good. Some instructors may want it to include proofs of things like existence and uniqueness, but I'd say the author made sound choices of what to omit and what to include.
I didn't notice anything culturally sensitive.
This is a good book for the intended course, but I think most students would want it printed.
Table of Contents
Chapter 1: Introduction
Chapter 2: First Order Equations
Chapter 3: Numerical Methods
Chapter 4: Applications of First Order Equations
Chapter 5: Linear Second Order Equations
Chapter 6: Applications of Linear Second Order Equations
Chapter 7: Series Solutions of Linear Second Order Equations
Chapter 8: Laplace Transforms
Chapter 9: Linear Higher Order Equations
Chapter 10: Linear Systems of Differential Equations
Chapter 11: Boundary Value Problems and Fourier Expansions
Chapter 12: Fourier Solutions of Partial Differential Equations
Chapter 13: Boundary Value Problems for Second Order Linear Equations
About the Book
Elementary Differential Equations with Boundary Value Problems is written for students in science, engineering, and mathematics who have completed calculus through partial differentiation.
- An elementary text should be written so the student can read it with comprehension without too much pain. I have tried to put myself in the student’s place, and have chosen to err on the side of too much detail rather than not enough.
- An elementary text can’t be better than its exercises. This text includes 1695 numbered exercises, many with several parts. They range in difficulty from routine to very challenging.
- An elementary text should be written in an informal but mathematically accurate way, illustrated by appropriate graphics. I have tried to formulate mathematical concepts succinctly in language that students can understand. I have minimized the number of explicitly stated theorems and definitions, preferring to deal with concepts in a more conversational way, copiously illustrated by 250 completely worked out examples. Where appropriate, concepts and results are depicted in 144 figures.
Although I believe that the computer is an immensely valuable tool for learning, doing, and writing mathematics, the selection and treatment of topics in this text reflects my pedagogical orientation along traditional lines. However, I have incorporated what I believe to be the best use of modern technology, so you can select the level of technology that you want to include in your course. The text includes 336 exercises – identified by the symbols C and C/G – that call for graphics or computation and graphics. There are also 73 laboratory exercises – identified by L – that require extensive use of technology. In addition, several sections include informal advice on the use of technology. If you prefer not to emphasize technology, simply ignore these exercises and the advice.
About the Contributors
William F. Trench, PhD. Andrew G. Cowles Distinguished Professor, Trinity University (Retired).