Author: Lewis, Andrew D.
Date of Publication: 23 January 2017
Title: Introduction to Differential Equations for Smart KidsLL
Location: https://mast.queensu.ca/~andrew/teaching/pdf/237-notes.pdf . Accessed 18 June 2019.
His opening lines in the Preface tell you what is ahead and why,
"This book is intended to suggest a revision of the way in which the first course in differential equations is delivered to students, normally in their second year of university. This course has traditionally grown as an offshoot of the calculus courses taught in the first year, where students often learn some techniques and tricks for solving specific problems, e.g., for computing specific derivatives and integrals. This is not an entirely unreasonable thing to do, since it is difficult to imagine being able to practice mathematics without being able to handle calculus with some of the basic special functions one encounters in a first course.
"Moreover, a first calculus course often comes complete with many insights into the meaning of, and uses of, differentiation and integration. However, this `techniques and tricks' method becomes less valuable for ordinary differential equations. The fact is that there are very few differential equations that can be solved, and those that can be solved only succumb after quite a lot of work.
"Thus, while I do believe it is essential to be able to solve a number of differential equations “by inspection”—and I expect students taking the course for which this is the text to be able to do this—the proliferation of computer packages to carry out efficiently and effectively the tedious computations typically learned in a differential equations course make one reconsider why we teach students multiple ways to solve the same small set of differential equations.
"This text is the result of my own reconsideration of the traditional first course in differential equations. As an instructor, the question becomes, `If I do not teach all of the usual techniques and tricks for solving differential equations, what do I replace it with?'
"My choices for answers to this question are the following
"1. Make sure students know that differential equations arise naturally in a wide variety of fields, including the sciences, engineering, and the social sciences.
2. Make sure students know what a differential equation is.
3. Appreciate how to use a computer when working with differential equations.
4. Understand the character of solutions, rather than just producing their closed-form expressions.
5. Introduce transform methods for differential equations, since these are very powerful."
These are all Professor Lewis' words. Here are our words.
The text is hot-linked so it is somewhat easy to bounce around (I wish there were links BACK to the Table of Contents now and then in the text) in the pdf file and the Table of Contents are rich and detailed to make the reader want to do just that. The diagrams are good, appropriate, and helpful.
First things first. The author offers example after example with context in which differential equations have a major role in formulation and then he goes on to give the essential vocabulary the reader will need to unlock these treasures.
The work is actually quite a bit more about theory than application, but the essential theory is nicely done in concise notation AND there are proofs galore. to satisfy any theoretically inclined approach in existence, stability, convergence, etc, More than we would ever want, but there for the taking. You could send students to this text for the proofs of important results on which we base our teaching. The author presents road maps and overviews all the way through the development so one can keep track of why we are doing this or that and how it links to upcoming and just been there.
There are exercises which are of the theoretical approach in applying the theory to examples and moving details of the theory along. No applications appear in exercises, though.
So, while the text begins with great promise of applications it really is about supportive theory, wonderfully rich supportive theory. If you want your students to really see what it takes to support what you do in your normal coursework then this is a good place to take them for a while and show them what is really needed so they can actually DO things with differential equations.
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