Learning Math Differently: Understanding ODEs

[animboy]

Hi,

Usually, it takes a while for me to digest information, because I have a lot of filters in my mind and to remember and understand things I have to put all the new information in context. I have to have an interpretation of the content. For this reasons I am doing terribly in my ODE course because the lecturer just spurts out case specific methods for solving ODEs, one after another. It feels a bit like I am learning chess instead of Math.

I am very depressed that I do not fully understand differential equations, why do they exist in the current scheme of mathematics? How are they different from other regular equations? As far as I have thought, ODEs are alternate representations of normal functions, but are somehow more useful in practical situations, why would that be?

For a more specific question, consider a first order linear separable ODE, why do we bring bring the g(y) term to the denominator on the left hand side, what are we trying to do by separating the equation in this manner? If your answer is, "because then you can answer it", I already know that, but I need a graphical or geometric interpretation to convince me that that is the only way to do it. I suppose it's a pretty hard request, since most people do not go about learning in this way.

 

[Angry Citizen]

How are they different from other regular equations?

Differential equations relate rates to other functions. These rates are derivatives. Whereas 'regular' equations relate an independent variable to a dependent variable through a function, differential equations exist to relate a function to itself. The solution to a 'regular' equation was an independent variable that satisfied the conditions brought about by the dependent variable. The solution to a differential equation is a set of functions (differentiated by a constant, unless initial or boundary conditions are specified) that satisfy the relation of the rates of the function to each other. You're finding functions now, not numbers.

For a more specific question, consider a first order linear separable ODE, why do we bring bring the g(y) term to the denominator on the left hand side, what are we trying to do by separating the equation in this manner? If your answer is, "because then you can answer it", I already know that

Well, that's it. An ODE class at your level is intended to tell you how to find a function given a relation consisting of its derivatives (and possibly the function itself as well). Separable equations exist because the integral of dx/y is just going to be x/y, because when you integrate with respect to a variable, other variables are treated as constants. But this gives an incorrect solution, because y is not constant. You have to transfer the y over to the side with a dy, and the x over to the side with a dx. Then you are integrating with respect to consistent variables, and the solution of dx/x is ln(x) - very different from x/y!

Others may be able to give a more rigorous explanation.

 

[mege]

Are you in a DiffEq for Engineers (generally 2cr) or a 'full' ODE course (generally 3-4cr)?

Also, something to note - most 'easilly solvable' differential equations generally don't have an easy graphical representation (hence the rigorous methods to solve them). The only (nice) graphical representations of solvable differential equations that I can think of are population models and some simple mixture problems.

With seperable differential equations - think of it this way: you need to get like terms together. That's really all it is: dx and dy are just another term in the equations as opposed to strictly an operator like they're treated in earlier classes.

 

[homeomorphic]

Well, I seem to be lacking in energy today, so I won't be able to give the best answer, but let me make a few comments. I hated my undergraduate ODE class and felt similarly.

I think of ODEs as just vector fields. This picture isn't always useful for actually solving specific equations, but it is very helpful for understanding the general ideas and can give you helpful qualitative information. When you solve an ODE, you are just flowing along some vector field. You just go where the arrows point.

One book on ODEs that emphasizes this viewpoint is the ODE book by V.I. Arnold. It really gets into the theory. It's somewhat advanced and difficult by undergraduate standards, but it might help to take a look at it if you can find it at the library.

Couple other comments. You need some linear algebra to understand linear systems. To understand them completely, you need some sort of advanced linear algebra (Jordan canonical forms). Once you have seen that, it is possible to make sense of some of the disorderly non-sense and cheap tricks that you see in undergraduate ODE classes, like all those seemingly arbitrary factors of t that come up when you have eigenvalues with multiplicity.

Another useful picture for 2nd order ODEs can be found somewhere in Feynman's lectures on physics, volume I, where he discusses the harmonic oscillator. This gives a lot of insight into second order linear ODEs with constant coefficients. Basically, you want to imagine something moving around in a circle. So, that's really where those cosine and sin terms come in. That's what happens when you have complex eigenvalues. It helps to have a really good grasp of complex numbers, here, too, for which the first chapter of Visual Complex Analysis would be very helpful.

Also, exact equations ought to be understood either in terms of div, grad and all that, or in terms of exterior calculus, which is another way of doing the same stuff.

So, there you have it. I have outlined a cure to the plague that is the undergraduate ODE class. Perhaps, there are still a few nasty things I didn't address, but you can't win 'em all, I guess.

 

[Lavabug]

I felt pretty much this same, it just felt like doing mindless algebra + applying some "recipes" to get the solution. I would just bare with the resolution methods and worry more about interpreting the solution (if you are a physics or engineering major). Ie: solutions with real valued exponentials describe hyperbolic behavior, like the diffusion of a gas or heat. The solution for the homogenous part of a DE with complex eigenvalues describes something that evolves periodically like an oscillator, which can originate from a source like an initial kick or forced resonance, given by the particular solution (RHS).

Why are type A differential eqns always solvable by such and such method? Some bright mathematician before my time figured it out. Why are solutions to PDE's expressable as a product solution of separate functions dependent on separate variables? Still haunts me to this day, but I think fitting all that into a science/engineering curriculum would be practically impossible.

 

[homeomorphic]

I would just bare with the resolution methods and worry more about interpreting the solution (if you are a physics or engineering major).

In doing so, you have to realize that you would be missing out on some great insights as to why it all works and the meaning and beauty of it all. Perhaps, under such trying circumstances, some compromise is best, but I don't think it's wise to just completely accept what you are taught unquestioningly. Better to make some effort to understand things more deeply. Otherwise, we just perpetuate the problem.

Why are solutions to PDE's expressable as a product solution of separate functions dependent on separate variables?

I would tend to think of that as really arising from Fourier series, rather than the other way around that you see in many books. The first step is to assume that it's a product of functions in separate variables. But, I think it's only in hindsight that you get that idea, after actually trying to work with Fourier series a lot practice using them to solve the heat equation or wave equation. Not just solve the equations but try to get some intuition for why it's working. If you put give the heat equation a sine wave (function of one position variable), it behaves in a particular way, since the sine wave is an eigenvector of the differential operator. So, you get the idea of building up any solution in terms of sines and cosines because those are the easier to understand. Well, I'm a topologist, so it's not really my area, and I haven't thought it through completely, but I think the answer is lurking there.

Still haunts me to this day, but I think fitting all that into a science/engineering curriculum would be practically impossible.

I don't think it would be impossible to do considerably better than what is being done now. In fact, it would be trivial, assuming no one started whining about being asked to actually understand what they are doing. But change comes slowly sometimes.

It's the silly taboo against talking about intuition and visualizations. If only people were a little bit less rigidly formal, many of these problems would be instantly fixed.

 

[Lavabug]

In doing so, you have to realize that you would be missing out on some great insights as to why it all works and the meaning and beauty of it all. Perhaps, under such trying circumstances, some compromise is best, but I don't think it's wise to just completely accept what you are taught unquestioningly. Better to make some effort to understand things more deeply. Otherwise, we just perpetuate the problem.I would tend to think of that as really arising from Fourier series, rather than the other way around that you see in many books. The first step is to assume that it's a product of functions in separate variables. But, I think it's only in hindsight that you get that idea, after actually trying to work with Fourier series a lot practice using them to solve the heat equation or wave equation. Not just solve the equations but try to get some intuition for why it's working. If you put give the heat equation a sine wave (function of one position variable), it behaves in a particular way, since the sine wave is an eigenvector of the differential operator. So, you get the idea of building up any solution in terms of sines and cosines because those are the easier to understand. Well, I'm a topologist, so it's not really my area, and I haven't thought it through completely, but I think the answer is lurking there.
I don't think it would be impossible to do considerably better than what is being done now. In fact, it would be trivial, assuming no one started whining about being asked to actually understand what they are doing. But change comes slowly sometimes.

It's the silly taboo against talking about intuition and visualizations. If only people were a little bit less rigidly formal, many of these problems would be instantly fixed.

I do agree in principle with what everything you are saying. I may have had the curiosity in pure math beaten out of me with all the heavy courseloads and everything. I really had a lot of qualitative questions in my ODE and PDE/fourier series/transforms courses, but the pace of said courses really discouraged from delving into those types of questions, which saddened me. Same goes for some physics courses like QM, where 2 weeks into the course I'm expected to know what CSCO or a tensor product is by looking up their "proofs" in a high level reference book.

I fully agree with rigor often being an obstacle in learning. I hope it didn't seem like I was encouraging a non-understanding of ODE's, I was just sharing my experience with them and how I coped with it (which in hindsight was probably not helpful to the OP at all).

 

[animboy]

Alright, you said "differential equations exist to relate a function to itself", this is what I needed. I understand the fundamental difference now. I can build on this myself now, also I will check out all the topics recommended by homeomorphic. Better sooner than later. I don't plan on going on like this, I will try to get to sort this out its better that way in the long run.

 

[homeomorphic]

I can build on this myself now, also I will check out all the topics recommended by homeomorphic.

Well, it could be a lot of work, so I'll give you a heads up on the most difficult parts.

A lot of things could be explained at a lower level than Arnold's book, but it's the only reference I am familiar with. Any time I look at any ODE book that is "elementary" ODEs, the same ugly problems come up, so Arnold is the best option I know of. The prerequisite to Arnold's book is a little bit of real analysis and maybe some comfort level with proofs (that's a year long, difficult math class). However, I think, you might still be able to gain something from it.

Mark Meerschaert's book on mathematical modeling covers some ODE in a nice way, I think, with less prerequisites. I actually took the author's class, but I'm not sure how closely we followed the book, and if what I remember from the class is all there.

The other topic that could be kind of hard is Jordan forms of matrices. You can probably look it up on wikipedia. I would skip the proof for the time being, but if you're interested, eventually, I would recommend Linear Algebra Done Right for the proof.

The rest of what I mentioned shouldn't pose too big of a problem.

 

[Jorriss]

I think the best basic ODE's book is Differential Equations by Sheldon Ross. I absolutely recommend it. He motivates theory as well as application. It's slightly above a standard LD course in differential equations but it's by no means advanced UD.

https://www.amazon.com/dp/0471032948/?tag=pfamazon01-20

 

[homeomorphic]

Also, for linear systems ODEs, it's good to just pick up any linear algebra book and start working through it. Eigenvalues and eigenvectors are the main thing to know. Just knowing the basics of that will cover a lot cases. But for the full story, you need Jordan forms, which is somewhat involved to understand in detail.