Talk:Differential Equations
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The Wikibook for Calculus already covers some portion of Diffrential equations, including a rather complete introduction to ODE. I think it would save time and effort to expand on that section, and make DE a part of that book. Over a period of time, other higher level books would also be integrated into calucus (such as series, complex analysis,etc).
Sorry for not reading this sooner, I was on vacation for the 4th. I'm not sure I agree this should be part of another book, although I'm willing to discuss it further. DE is a rather large and complex topic, easily a college semester even with barely touching partials. I think adding it into one giant book would be doing a disservice to both subjects- it would make the calculus book so huge as to be daunting to readers, and make the DE part a side note when it deserves more attention. I guess my mindset is more of the Unix philosophy as applied to books- do one thing, and do it well. Tieing things together into courses of study (calc, then DE, then linear algebra (or whatever order)) is the job of a higher order layer.
I took a quick glance at the DE section of the Calculus book. Its well written and covers a lot of ground. The problem I have with it is that it doesn't *teach* DE. Its a problem I have with a lot of wikibooks. Its a good reminder, if I wanted to review the subject it would work well. But if I had never solved a DE before and was studying the subject for the first time, I would be utterly lost. It has few examples, no practice problems and answers, little explanation of why things worked, no discussing of the theory behind the methods, nothing of the practical applications. It reads more like an encyclopedia entry than a book. Which makes sense, given the nature of wikibooks springing from wikipedia. And its better than the average wikibook out there, having read many that looked like powerpoint slides. But still of low use as a classroom or self teaching aid.
Which brings us to another issue on adding this DE to the calculus book. I don't want the book to look like the standard wikipedia style. I don't like the standard style, being too much encyclopedia and not enough book. Given that I won't be using the same style, I'm not sure that it would be a good idea to assimilate the content into one book. Within a book, IMHO, one ought to make an attempt to keep the style of the information the same. To do otherwise is confusing. And while I have no doubt my style will have its own deficiencies and I'm happy to listen for impovements, I doubt you'll be able to convince me to switch to standard wikibook style.
--Gabe Sechan July 5, 2005 22:01 (UTC)
Just laying out some ideas for a brief reformation, im going to write some things on the formation of differential equations before the tackling of the concept of a solution so as to provide what I hope will be a more intuitive and understanding approach.Robert Carr 03:38, 18 January 2006 (UTC)
It's a shame to see this book neglected -- it would be a really nice resource to have an online book on DEs. I think I'll write some content for it. I've already written Variation of Parameters (under Constant Coefficients) and maybe I'll also add something about Laplace transforms, which are really fun and powerful. Perhaps I'll even do a section on series solutions if I'm feeling bold.... N Shar 05:17, 31 January 2006 (UTC)
Please do. I'm sorry the book has become neglected as well- I just don't have time to add to it these days. And on top of that, I lack the confidence in some of the more advanced material- I am not a math professor. As the material gets more advanced I felt more like I was regurgitating source books than I was writing from my understanding of the material. --Gabe Sechan 19:22, 31 January 2006 (UTC)
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[edit] Relationship of this book with the Wikipedia entry on Ordinary Differential Equations
Reading the current version of the Wikipedia entry on ordinary differential equations, it seems to me that much of that material belongs in a textbook, rather than an encyclopedia entry. I think the encyclopedia should have definitions and motivation for studying the material. I have written a comment to that effect on the wikipedia article, and I'm in the process of writing some of that material for the wikipedia article.
Now as far as the wikibook is concerned, I think the title shhould be "Ordinary Differential Equations", because that describes the content. My background is in partial differential equations, so I'd like to advance that side of the wiki enterprise, but I think ODE comes first logically. I'm now retired with some time on my hands, so I'd like to help with any or all of these enterprises. So far I haven't been able to get a wikibooks ID, but my wikipedia ID is "donludwig". Email is ludwig@math.ubc.ca 154.5.94.9 17:46, 23 February 2006 (UTC)
[edit] Book Development
I'm currently studying Diff EQ and would like to contribute to this book. This is the second time I've taken this course, and I feel that there are many students out there that would benifit from a free resource that could explain topics from a non-textbook-writer point of view. More importantly, I'd like to expand the example problems to cover as many examples as possible since this is the main way of learning any subject: practice. Please contact me so that we can discuss further development at curtwulf at gmail dot com. Thanks. Curtwulf 12:23, 29 March 2006 (UTC)
[edit] Definitions
Arnold defines an ODE as follows. An ODE is a process evolving in time that has the following 3 characteristics: (1) deterministic (2) finite dimensional (3) differentiable. where deterministic means that the entire past and future of a process is determined by its initial value. finite dimensional means that the set of all possible states of your process (the phase space) can locally be parametrized by finite many real numbers (eg.: a sphere can locally be parametrized by two real numbers, so does the plane, the real line, etc). Differentiability means that the time evolution processes can be described by an equation that is differentible.
I like this approach to ODE's by Arnold better then the one normaly found at books because it is more conceptual.
Response: With all due respect to Arnold, that definition is too narrow for many practical purposes. It focuses on "processes evolving in time" which is not the only situation where ODEs arise. For example, it excludes boundary value problems in which case the description "the entire past and future of a process is determined by its initial value" makes no sense.
Furthermore, the requirement (3) is too restrictive. Consider, for instance, the differential equation y'(t) + y(t) = f(t) where f is a step function. Such equations arise all the time in electronic circuits; the step in f corresponds to throwing a switch. The solution y is continuous but not differentiable. Of course, to make sense of the differential equation, we need to view it as an integral equation, as in:
y(t) = y(0) + int0t [ f(t) - y(t) ] dt
which is what one does when proving the existence and uniqueness theorems anyway.
-- rr
[edit] Reformat
I'd like to create a standard layout for accessing the examples, problems, and answers, like at the end of the page explaning the topic so the confusion in the chapter on First Order Equations doesn't happen.
