Trench, William F. 2014. Elementary Differential Equation. Books and Monographs.

By Brian Winkel


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Author: Trench, William F.


Date of Publication:  2014.


Title: Elementary Differential Equation. Books and Monographs


Accessed 16 August 2016.

Elementary Differential Equation. Books and Monographs.   Book 8. .

Elementary Differential Equations with Boundary Value Problems. Books and Monographs. Book 9.  Accessed 1 May 2014. .

Both texts are also available at Scholar Commons of the University of South Florida Textbook Collections. .

Both texts also available as Approved Textbooks at the American Institute of Mathematics (AIM)  at .

This is the most widely cited and referenced free online text for differential equations and is very complete.

Annotation:  These two books are exactly what they say they are. Moreover, they are FREE. Go to the Digital Commons site at Trinity University, San Antonio TX USA ( ) and you can freely download either version and a number of other texts.  The author has gone to great lengths to offer a reasonable level of theory for first course, a great number of worked examples and illustrations of techniques and applications, and a good number of interesting applications problems grouped at the end of reasonable sections occur in the first two general areas of study for ordinary differential equations, e.g., first order and second order ordinary differential equations. While there are scattered about a few application type exercises throughout the text beyond these first two set of applications, the text is mostly a rich treatise and an informal style conversation with the reader about the techniques and usefulness of the various solution strategies of ordinary (and in the case of Book 9 partial differential equations). These texts have had years of use through commercial publishers and the author has now made them freely available to the public in this Open Access venue. It is interesting to note that these books are very high in the Top 10 Faculty Downloads but are not in the Top 10 Student Downloads. Nevertheless, we found the texts to be very comprehensive, very readable, engaging, and mathematically correct and complete.  We believe a teacher could send a student to a specific section, ask them to read the material, follow the examples, and submit some of the exercises as evidence of command of the technique, perhaps to be followed up by an exam or quiz question.

In the Preface to Book 9 the author says

• 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 2041 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 299 completely worked out examples. Where appropriate, concepts and results are depicted in 188 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.

All the formulae one could want are offered, and in most cases, derived in these texts.  In the sections on Numerical methods there are rich illustrations with tables and comparisons of methods to motivate the value and need for numerical approaches.  The Heat Equation is just presented with no derivation or intuitive motivation while the Wave Equation is thoroughly presented and motivated.

Further, with respect to technology, in the section on Fourier Series of Book 9, p. 599 the author says,


“The computation of Fourier coefficients will be tedious in many of the exercises in this chapter and the next. To learn the technique, we recommend that you do some exercises in each section `by hand,’ perhaps using the table of integrals at the front of the book.  However, we encourage you to use your favorite symbolic computation software in the more difficult problems.”


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Researchers should cite this work as follows:

  • Brian Winkel (2016), "Trench, William F. 2014. Elementary Differential Equation. Books and Monographs.,"

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