# Elementary Differential Equations with Boundary Value Problems

William F. Trench, Trinity University

Copyright Year: 2013

Publisher: A.T. Still University

Language: English

## Conditions of Use

Attribution-NonCommercial-ShareAlike

CC BY-NC-SA

## Reviews

I have used this book as the main textbook in the differential equation course in Spring semester 2020. Overall, the book consists of well-organized chapters and connects with each chapter smoothly. I think it is a good textbook when teaching the... read more

I have used this book as the main textbook in the differential equation course in Spring semester 2020. Overall, the book consists of well-organized chapters and connects with each chapter smoothly. I think it is a good textbook when teaching the undergraduate level course, ordinary differential equations during one semester. Some who want to cover partial differential equations chapters may need one more semester.

I could not find the error in the contents when reading the book.

Overall, the contents fit the undergraduate students. But some explanations look substant and so the instructor needs to choose the necessary parts to make more effective lectures.

The author's description is clear and makes the student understand easily.

Almost all are correct and concise. But, some terminologies made students confused because it mixed the concepts repeatedly using the same terminology. I expect it would be updated in the future.

Every content was clear and transparent. But I thought it would be much better if several more techniques using Laplace Transform was added in the Chapter 10 Linear Systems of Differential Equations.

The text book contains necessary contents students should know when taking the differential equation courses. But, instructor should focus on 8-9 chapters if they want to complete the course only one semester.

I do not know exactly but I felt the text size was small and the spacing was a little bit narrow. I think it could be updated when modifying the platform.

I could not find the grammatical errors.

Usually, the math books are neutral and do not have personal biased thoughts. Of course, I could not find it.

I liked to use the book when teaching the differential equation course. The solution key was so helpful to create my answer key for homework, quizzes, and tests. Students also felt convenient because they could check their answers immediately.

This is a great book for a 1 quarter term (10 week) class on introductory ODE theory and application. I appreciate the complete index - an index is often lacking in an OER but there is a useful one in this book, along with an accurate TOC. read more

This is a great book for a 1 quarter term (10 week) class on introductory ODE theory and application. I appreciate the complete index - an index is often lacking in an OER but there is a useful one in this book, along with an accurate TOC.

All content I read is error free. I suppose I will not truly know this until I use it in my course. :)

This textbook has all topics normally covered in a one term ODE course. It was originally published 7 years ago so I imagine the errors have been corrected.

Great explanations. I appreciate the way that some examples are included seamlessly as part of the text and others are called out by an easily understandable numbering system that includes the section number and example number.

I have not found any inconsistencies. The ordering of the topics is straightforward and similar to other ODE textbooks.

I believe this text was originally published by a "typical" textbook publisher and so has a very detailed level of organization. This is extremely helpful for students as well as for faculty in their planning of the course coverage.

As mentioned above, a typical organization of topics that can be rearranged if desired. I will likely teach "as is" the first time and then make my own adjustments as needed.

The pdf version contains clickable links in the textbook that jump to the page where there is a specific image or example. This is a level of "interactiveness" that I have not seen in many OERs and I really like this feature. The presentation of the images is great - no issues.

I have not noticed any grammatical errors thus far.

Again, I have not noticed any issues on this front. This book is not one that includes examples with race, ethnicity, and background featuring in the examples.

I am excited to try this book in Fall 2020. It seems to be the most organized OER I have investigated and I think that it will be easy to use and easy for students to follow. I also plan on using an associated MyOpenMath homework course to go with the Trench textbook.

This textbook contains all of the topics that are usually covered in an undergraduate course on ordinary differential equations and has a section devoted to partial differential equations of the second order. The text does satisfy the requirements... read more

This textbook contains all of the topics that are usually covered in an undergraduate course on ordinary differential equations and has a section devoted to partial differential equations of the second order. The text does satisfy the requirements of the Ohio transfer module. It can be used for a course targeting math majors as well as a course on elementary differential equations for engineering majors. Comparison of different methods and their limitations provided in the text is very useful. It is a fine quality textbook. There are only two topics that I would add or modify. (1) I would add a discussion of the application of the Laplace transform to the higher order equations and systems. In the current edition it seems to be covered only for the equations of the second order due to the placement of the section on the Laplace transform. (2) I would add the concept of the equilibria and their stability for the first order autonomous equations as well as introduce traditional terminology for the equilibria in two-dimensional systems (saddles, nodes, ...) and a discussion of their stability.

This text appears to be accurate, well-written, and error-free.

This textbook contains a solid foundation in differential equations for students of mathematics, physics, and engineering. The level of rigor is appropriate for an undergraduate audience.

The terminology that is used in the text is explained and the definitions are clear. A topic of equilibrium solutions and their stability are explained without introducing common technical terminology which is an omission, in my opinion. Overall, the explanations are not on the easier side: a reasonable level of effort is expected from the reader. The author does not make complicated topics seem elementary. Important details are not omitted; they are pointed out and discussed.

The author uses consistent notation and vocabulary.

The material is presented in sections of reasonable length each focused on a well defined topic. After each section there is a problem set on a content of the associate. Answers to a selection of problems are provided at the end of the book.

The order of the material differs from the order of the material in some of the popular textbooks on the topic. For example, higher order equations do not immediately follow the second order systems but come after the series solutions and the Laplace transform. Overall, the order is reasonable.

I did not notice any interface issues. The presentation and typesetting of formulas and equations is well-done. The figures and images are clear and conveniently placed.

I found no grammatical errors.

This is a textbook on basic differential equations. There is not cultural context to discuss.

Most of the content is present, and appropriately indexed. Its use would depend heavily on the content covered at your institution. I could not find Lotka-Volterra equations (Predator-Prey), which is a significant topic in our Differential... read more

Most of the content is present, and appropriately indexed. Its use would depend heavily on the content covered at your institution. I could not find Lotka-Volterra equations (Predator-Prey), which is a significant topic in our Differential Equations course. Bifurcations and Lorenz equations were limited and missing, respectively. These were less important topics, but still missing. The content of Chapter 10, Linear Systems of Differential Equations, seemed too compressed and may benefit from expansion to multiple chapters. This is a large topic in Differential Equations that would benefit from more examples and explanation. I agree with another reviewer that, although the author is writing for the students, there are a lot of skipped steps and language where a student at the introductory-level would struggle to follow along.

No content inaccuracies were found. Some things were presented in a way that is not standard, but many of those were positively connecting to other material in the course. I found it refreshing.

The material presented is consistent with differential equations, and is not going to become obsolete. Any necessary updates appear to be straightforward to implement.

Most of the material is presented very well. The language is clear and accessible, and the mathematical content is appropriate for the variety of levels entering differential equations students have. Several sections have summary sections outlining the processes of the different techniques, which seem very student-friendly. However, there are also sections that are particularly short, or not articulated as clearly (Ch. 10 in particular).

The author has been mostly consistent in language and framework. There are some sections that do not contain the summary sections that clearly articulate the techniques, but the style is similar otherwise. The blue text is used for multiple things (links, theorems, summaries), and may be confusing to the reader.

The text appears modular. Some of the solutions and links in the index may not be modular, but the text itself appears easy to subdivide and rearrange as needed for teaching the course. The authors choice to use colored subsection titles may prove to not be easy to subdivide, but without attempting this myself I cannot say.

Most of the topics are presented in a clear and logical fashion. Personally, there appears no reason for the ordering of chapters 3, Numerical Methods, and 4, Applications of First Order Equations. Since the topics of chapter 4 use the material in chapter 2, it would seem more logical to flip these chapters, or integrate the applications with the techniques in chapter 2.

The navigation of the links in the index and in the actual text appear to be functional. The images and other displays seem appropriate.

I did not spot grammatical errors in the text. However, there are several typos in the Table of Contents. There were not many errors spotted in the actual text, but when the table of contents has errors, that appears very grim to the reader.

I found no cultural biases in the content.

Much of this book is great, but it is unfortunately not sufficient for our first-semester differential equations course. The text would need modifications to include all the appropriate content and applications. Personally, the colored subsections (in red) make the subsections blend into the rest of the text. This may be a problem if printed, and may be difficult for color-blind readers. Additionally, the first chapter has few problems compared with the later sections, and it seems that would be a good place for many problems, as it is the students' introduction to the content. The errors seem particularly confusing in the table of contents, and I would like to see those fixed. Despite all those comments, this author has presented several concepts in an enlightening manner that I think will be beneficial to students. Variation of parameters and integrating factor in particular. I would certainly consider modifying this text with attribution to use sections of it in a differential equations course. There are a lot of good problems throughout the text, and good examples.

With 13 chapters covering standard topics of elementary differential equations and boundary value problems, this book contains all materials you need for a first course in differential equations. Given the length of the book with 797 pages, the... read more

With 13 chapters covering standard topics of elementary differential equations and boundary value problems, this book contains all materials you need for a first course in differential equations. Given the length of the book with 797 pages, the instructor must select topics from the book for his/her course. I believe that materials from Chapter 1 to Chapter 10, except possibly Chapter 7, should be covered in the course, while the last three chapters could serve as additional reading materials for students.

The book was very carefully written. I did not find any inaccuracy in the book.

As the book covers standard topics of differential equations and boundary value problems, the contents of the book will continue to serve students for a long period of time. Without specifying what technologies should be used, the instructor can choose the computer software to use in his/her course. This is great for the future use of the book given that technology is rapidly changing.

This book was carefully written. Each topic is accompanied by several well-chosen examples.

The notation and terminology are used consistently throughout the book.

Topics of the book are divided into small sections of appropriate lengths. A brief overview of the topics to be covered in each chapter is presented at the beginning of each chapter. This is very convenient for both instructors and students to read.

The book is very well-organized.

The hyperlink navigation in the textbook is very helpful.

The book was very carefully written. I did not find any grammatical error in the book.

As a mathematics book, this book contains no cultural relevance issue.

I highly recommend this book for a first course in differential equations. It is also an excellent choice for a sequence of two courses in introductory differential equations and partial differential equations. I read another textbook by this author and I like both of the books.

I'm going to hedge this review by confessing that I am not a specialist, either by training or by inclination, in differential equations. I know the subject well enough to teach it well to undergraduates, but I don't consider myself to say... read more

I'm going to hedge this review by confessing that I am not a specialist, either by training or by inclination, in differential equations. I know the subject well enough to teach it well to undergraduates, but I don't consider myself to say authoritatively that any textbook of differential equations, in particular, this one, prepares the student well for further studies. So, the reason I give it a 4, instead of a 5, is that 1 is the degree of my own uncertainty about its comprehensiveness.

I've taught introduction to differential equations using this text twice now. Both times I covered at least the first 5 chapters. Among that material I found no errors. I imagine that the rest of the book is as tight.

Again, I do not specialize in differential equations, so, take my take on this with at least a grain of salt. But as far as I can tell, any math major who learns differential equations from Professor Trench's text will be well prepared for the mathematics which comes next.

My students, and I , have found this book to be both easy to comprehend and interesting.

I found no contradictions.

I think this book is organized as well as or better than any other text I've used in my nearly 30 years of teaching. This is not to say, however, that it could not be improved. There is a tension between organizing the introduction to a subject into bins based on its parts similarities and organizing it into parts based on what one can cover in a class meeting or two. I think this text favors the former over the latter. This may or may not be right for you. Choose accordingly.

The subject of each section builds on those of the previous, and in a natural way. Good job.

No complaints.

I noticed no grammatical errors in the first six chapters. There may be some, but they are not glaring.

Examples are not culturally insensitive. On the contrary. I've taught a diverse group of students from this text and have received no complaints. They may exist, but I don't think they are glaring.

Good text. If you, like me, are working at a college which serves, principally, poor students, this is text is a good choice. It covers what they need to know for future courses, it offers interesting supplemental material and the exercises represent a wide range of difficulty. Next time I teach Diff Eq, I'll, again, be using this text.

This a complete book for an introductory differential equations course. It has everything you want and more. read more

This a complete book for an introductory differential equations course. It has everything you want and more.

This is an excellent professional standard textbook. The author clearly developed this product with careful thought and meticulous detail.

The textbook does not highlight specific software, which I wish it did at times, but it also makes it timeless as the content will not age.

I think this is an excellent book with a narrative explanation style that makes things both clear and precise when need be. The author went to great lengths to use language relatable to students at this level. I found other textbook options were either too basic or too complex. This one is excellently written and by far my top choice.

Terminology is very consistent throughout. The author goes to great length to connect sections for smooth transitions.

The textbook is broken down in manageable pieces very well.

The author had a very clear flow and organization with a desire to integrate ideas as he goes along. Excellent textbook.

No Issues. Ebook even has hyperlinks to student solutions manual. Works very well.

Very Well Written.

The author knows his audience and has tailored it to them well.

One area not mentioned here that is relevant are homework exercises. There is an extensive set of exercises of vary degrees of difficulty with a comprehensive student solution manual.

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... 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.

n/a

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

### Author

**William F. Trench, PhD. **Andrew G. Cowles Distinguished Professor, Trinity University (Retired).