From: foltinek@math.utexas.edu (Kevin Foltinek) Subject: Re: differential equations Date: 26 Apr 2000 12:10:26 -0500 Newsgroups: sci.math Summary: [missing] In article <39064D35.9868227@wam.umd.edu> Oren Fromberg writes: > hi, I am an undergrad taking a one-semester course on differential > equations for engineers. I liken this class to a driver's education > course because I feel that I am simply learning to drive a car, instead > of how the car really works. I'm guessing, from your description, that the course is full of "tricks" like "if the equation is in this form, it's called a *** equation, and is solved by making a substitution of the form ???". And I'm guessing that, as a consequence, you don't see any "structure", as in, "how in the &&& would anybody think to do that?". If I'm right, then you're not alone. But here are a couple of points; one is a concrete suggestion, the other is a bit more abstract. (If my guess is wrong, the second point is probably still applicable.) First: find a book, such as Arnol'd's Ordinary Differential Equations, or something with a title like "Qualitative Methods of ODE"; these books tend to approach ODEs from a slightly different perspective. They discuss not so much tricks to solve them, but more general conceptual ideas about them, such as what you can say about the solutions when you can't construct them. Second: there is a general theory, which requires fairly sophisticated machinery, which explains where all of the "tricks" come from. There are methods that you can use to construct the tricks. Unfortunately, these methods typically take much longer than simply memorizing the tricks. A quite readable source is Olver's Applications of Lie Groups to Differential Equations (especially Section 2.5, I think). Olver doesn't quite explain the full theory, but I find his computational approach quite workable (as an introduction to what is in fact a rather rich subject). > Is there any advice out there that could > make me feel like I am learning something that is as beautiful as > algebra? The subject I mentioned in the second point is actually rather algebraic in nature (working with Lie groups and Lie algebras, and differential ideals); but it also requires some more background (a thorough understanding of manifolds, tensor analysis, differential forms) besides algebra. All of this, though, will take you far from your DE for E course, so remember to memorize all the tricks too! Kevin.