I'm looking for any resources you guys can point me to on "real" applications for differential equations. I have a relatively decent grasp of the math, but it's very frustrating for me learning sterile math with no idea what it's used for, especially since I figure that I'm going to actually have to use it down the road (aerospace engineering major). So far in the class we've done one (!) applications-type problem, something with a sprig, which the instructor basically gave up on after about 5 minutes, and said it would be extra credit for anyone who could solve it. I really think most of this is because the instructor isn't really that comfortable with Diff Eq, but it's his turn to teach it so we're SOL. I mean, for example, I get the idea that real-world diff eq involves a lot of computer applications - at least the book includes a lot of material aimed at it. Aside from his fumbling through the first section of that in the first chapter we've not touched a thing, because he has no idea how to use Maple. Of course our book (An Introduction to Differential Equations: Order and Chaos, Florin Diacu) does have some stuff in it, but IMO it's not very well written... it's a struggle just to interpret the basic concepts from it. Anyway, thanks for the help.
I'm not sure what you mean by "real" applications? If you are looking for practical applications, pick up just about any Physics book beyond introductory level. If you are referring to applications to problems that mimic reality in that they include friction, dissipation functions, and all the rest of the terms that can make Physics a very nasty subject, then you can perhaps look more toward advanced Engineering books. The problem with the term "reality" is that all physical models have some level of restriction on them that make the problem more tractable for a particular kind of solution process. Thus you won't find any problem trying to analyze "what really happens," if that's what you mean by "reality," because in a real kind of process there are too many different things to try to keep track of. So to do any kind of problem at all we need to assume certain processes in Nature (or manufacturing) are perfect. If it helps, most advanced courses will try to show you how to solve the specific types of problems that usually crop up in a particular field of study. For the sake of time and simplicity they take on one kind of problem at a time. It is usually left to the "interested student" to see what needs to be done if more than one of them are present in a given problem. That kind of learning is usually reserved for graduate students. I guess what I'm saying is that there IS a point to all of the "sterile math," it's just that you usually don't get to see it for a while. -Dan
hahah AE eh? wait til you get to numerical methods and CFd and finite element analysis and aero theory......you will see all those friendly-looking differential equations calling you to solve them. you 'll see that at that point, everything you have learned in your introductory class to ODE is basically useless. Go Maple!! Oh hand yes you will use the laplace transforms a lot in flight Dynamics and controls........you are warned!!
the o.d.e. book i like most for interesting applications is by martin braun. used copies are available cheap for a few bucks. i got mine for $2.95. there are applications to dating paintings, figuring why more sharks than food fish benefited from wartime cessation of fishing, theory of combat, stability theory. another i liked was guterman and nitecki where they show how electric circuits give rise to systems of linear equations.
ODE's are definitely used in circuit analysis. PDE's are VERY important, and the concepts of ODE's are needed to understand and solve that class of equations. A lot of real world problems are represented as PDE's, and you definitely need to know ODE's just like you need to understand calculus. Everything from fluid flow, heat equation, wave equations, maxwell's equations, etc... Anyways... If you don't click with your professor then watch these lectures: http://ocw.mit.edu/OcwWeb/Mathematics/18-03Spring2004/VideoLectures/index.htm Dr. Mattuck is awesome. Whenever I didn't grasp something in my lectures, I would turn here and usually with the combination of my professor and Dr. Mattucks it would make sense. My professor for diffeq was fantastic though. One lecture he modeled a college students kidney while out partying. He used the delta dirac function to represent the student taking a shot. It was very funny... and it's not everyday that you are seriously laughing in a math class.
i don't know about applications as i just started it, but the book by edouard gorsat on diff eq is excellent so far. i already learned something on page 2, and i just finished teachjing the cousre for 15 weeks out of standard books. Goursat starts backwards, he takes a family of functions f(x,y,r,s,t,..) considers thme as a family parametrized by r,s,t,... of impliocitly defiend functuioins y(x), and differentiates them to poroduce the diff eq they satisfy. this approach makes it clear that characterizing functions by diff eq's is natural. e.g. take the equation of a circle x^2 + ax + y^2 + by + c = 0. then differentiate it three times with respect to x, and eliminate a,b,c, and you get a single diff eq of third order. y'''(1+(y')^2) - 3y' (y'')^2 = 0. this is the diff eq satisifed by all lines and circles in the plane. each new equatioon lets you eliminate another avroiable, which makes it clear in hindsight why a diff eq of order n has an n parameter family of solutions. I never saw this before, and this stuff is on page 2 of goursat. a little later he is doing one parameter groups of transformations.