Ten lessons I wish I had learned before teaching differential equations (1997) [pdf] (opens in new tab)

Two days later he had an explicit solution based on two really non-obvious (bizarre) substitutions. I came away very, very impressed. The guy earned a 'we are indebted to ...' footnote in the paper.

I guess the point is someone has to come up with the tricks.

But the harder a subject, the longer it feels like learning a bag of tricks. The first partial differential equations course feels like you are working on only 3 problems for 4 months.

edit: I had a prof whose first DE course was at the graduate level. At the oral exam he was asked to give an example of a differential equation and all he could do was point to phi on the blackboard.

Many techniques in math are "tricks" like this. Think of solving a quadratic by completing the square, or integrating by substitution or integrating by partial fractions, etc. You could arrive at these techniques on your own, but that is a lot of trial-and-error, deep understanding of theory, and applying it, which all takes a huge amount of time. Meanwhile, previous mathematicians figured this out and we get to benefit from their work. ;)

Maybe your instructor didn't present it well - plopping out the answer without a good enough explanation of the technique, why it works, etc.

It took me a long time to return to learning higher maths, this time on my own according to the needs of my job, which is far less an ideal environment than University.

If ODE contains "a few basic concepts that they will remember for the rest of their lives" I never learned it, and I wonder what those concepts would be.

He acknowledges that concepts have prerequisites, and that certain topics are misleading or dead ends. You might even call the act of building a pedagogical scope and sequence for a topic graph traversal!

I wish I could have taken one of his classes. I wonder if there are other mathemetician/philosophers out there who have taken the problem of teaching as seriously as the math itself.

did anyone have an undergrad math education that was NOT like this?

What I didn't like about it was just all the memorization. I had no desire to memorize a bunch of formulas that I knew full well in the real world I'd look up in a table or type into a computer. What I wanted to learn how to do was solve problems using math, not memorize patterns of formulas to apply to problems.

So I didn't memorize them and instead went to work and earned money to pay for college. Still passed the class but it was one of my lowest grades. It's a hard class even without the memorization.

Hated sophomore ODEs.

Much later took a mathematical physics course that actually taught me ODEs. Also took a year of applied mathematics that really solidified PDEs and taught me complex analysis. Took another graduate course in calculus of variations that was useful. Another very senior-level physics course taught me greens functions, method of steepest descent, etc.

The Physics and Applied Math profs were vastly better at teaching Math than the Math profs. Only real problem was in the freshman Physics courses where they taught you sloppy vector calculus before you'd seen that from the freshman calculus courses (and generally if you tried to pick up Math directly from the Physics courses that were teaching concepts it was all sloppy physics math -- it worked but you never quite understood why...)

They are written in many ways by many different people.

But some of us have started to write books that are Free, in the sense that software is Free. I have a couple and in addition to making the text and the source available I also sell one of them on Amazon and it does OK (see http://joshua.smcvt.edu/linearalgebra), because lots of people prefer a paper version when they really get down to studying.

It's definitely something engineers care about, but not at the undergraduate level.

I didn't have a particular need for examples in a course, as differential equations were held up as the holy grail of math by my father, an optical engineer -- he used them at work fairly often, and frequently they were what made him better able to solve a problem than an engineer who wasn't comfortable using them.

Don't know if the professor could've done anything better or if they had no choice but to plow through it because of the time constraints.

This made me laugh; I think he wanted you to laugh too.

Some DE courses could be better described as _histories_ of the applications of mathematics. Consequently they appear to be little more than a patchwork of tricks and hints. My DE professor, as adept with applications as theory, weaved this patchwork together quite skillfully and held our interest.

BTW his DE skills paid his bills extremely well (oil companies have lots of DEs to solve), while the university position allowed time for theory and was frosting on the cake.

I'm very surprised this isn't standard material. It makes the parallel between Laplace and Fourier transforms so much more intuitive, because you get Taylor series as a parallel to Fourier series.

[1] https://youtu.be/zvbdoSeGAgI

And, if regard the material on differential equations as essentially nonsense, then good luck getting NSF grants for research in the subject!

Actually, can communicate a lot of good information in a course in differential equations, but to do this apparently need some exposure to some of the leading applications of differential equations.

That is where the wave equation comes in, in the first course on PDEs.

As a body of work Differential equations are so messy that theorists landed on so many disparate results. As such Differential equations courses are commonly taught as a "survey of the land" type of courses, so they tend to be incoherent. On the other hand if the teaching focused on practicality there is a lot of commonality among the practical cases.

I have an order of magnitude fewer docs that you; I can't imagine hand organizing that many!

http://pgbovine.net/publications.htm

this is something i try to do for all my papers, although of course people will still need to manually re-name to fit their own conventions. but at least it's better than Guo.pdf

I also use Green's functions, which are equations that describe the response of a medium to an impulse (think the propagation of sound waves from a source, though I do different stuff), by using convolution.

But I think jofer is the only other geophysicist on HN so we're probably not representative. Nonetheless, a lot of HNers have a physics, classical engineering or chemistry background and use similar tools... just not to find out what happened tens of millions of years ago.

"By the end of that summer of 1983, Richard had completed his analysis of the behavior of the router, and much to our surprise and amusement, he presented his answer in the form of a set of partial differential equations. To a physicist this may seem natural, but to a computer designer, treating a set of boolean circuits as a continuous, differentiable system is a bit strange. Feynman's router equations were in terms of variables representing continuous quantities such as "the average number of 1 bits in a message address." I was much more accustomed to seeing analysis in terms of inductive proof and case analysis than taking the derivative of "the number of 1's" with respect to time. Our discrete analysis said we needed seven buffers per chip; Feynman's equations suggested that we only needed five. We decided to play it safe and ignore Feynman.

The decision to ignore Feynman's analysis was made in September, but by next spring we were up against a wall. The chips that we had designed were slightly too big to manufacture and the only way to solve the problem was to cut the number of buffers per chip back to five. Since Feynman's equations claimed we could do this safely, his unconventional methods of analysis started looking better and better to us. We decided to go ahead and make the chips with the smaller number of buffers.

Fortunately, he was right. When we put together the chips the machine worked. The first program run on the machine in April of 1985 was Conway's game of Life."

I'm sure there are others, but this is a pretty good introduction. It focuses quite a bit on numerical solutions (using python programs that are automatically graded.)

So, what are they really? Well, the key things you can do with them are (1) "boring" linear algebra operations (you can add and subtract them, and multiply them by scalars) and (2) multiplying by some function and taking the integral. E.g., what delta(x) -- the Dirac delta function -- really is, is a thing such that when you compute integral f(x) delta(x) dx, you get f(0).

And so pure mathematicians have ways of dealing with them that make this property more explicit. The theory of distributions says: no, these aren't functions, they're linear functionals on the space of functions (e.g., the delta function is the thing that maps f to f(0)). So now you're no longer allowed to write them as integrals, which means that the very close analogy between "distributions" and ordinary functions is obscured, and e.g. if you need to do a change of variables you can no longer just do it the same way you already know about from doing integrals.

Alternatively, the theory of signed measures says: no, these aren't functions, they're kinda like probability distributions except that the total "weight" doesn't need to be 1 and the density can be negative in places. They are naturally applied not to points but to sets of points. (E.g., the delta function is the signed measure that gives a measure of 1 to any set including 0 and a measure of 0 to any other set.) Now you are allowed to write those integrals, but instead of writing integral f(x) delta(x) dx you need to write integral f(x) dH(x) where H(x) is the "Heaviside step function", so instead of delta(x) appearing there you have (morally) its integral, and again if you want to change variables or something you need to know a new set of rules for what you do to the measure.

Note: I have skated over some technicalities. They are quite important technicalities. Sorry about that.

The sloppy non-rigorous physicists' and engineers' notation, where you just pretend the damn thing is a function and manipulate it as you would any other function, is more convenient. (Right up to the point where you do some manipulation that is safe for actual functions but gives nonsense when applied to singular things like delta functions, and get the wrong answer.)

It's a little like calculus notation. The "Leibniz" notation we all use these days writes derivatives as dy/dx as if dx and dy were just small numbers (compare: we write integrals against distributions as integral f(x) delta(x) dx as if delta were just a function), which is kinda nonsensical if you take it too seriously but very convenient because it makes things like dz/dy dy/dx = dz/dx "obvious", which is not just coincidence but has something to do with the fact that derivatives really are kinda like quotients (in fact, they are limits of quotients). Similarly, using "function" notation for distributions lets you write things like "integral f(x) delta(x-3) dx" and see that "of course" that's f(3), and this convenience isn't mere coincidence but has something to do with the fact that distributions really are kinda like functions (and in fact every distribution "is" a limit of functions).

Newton had a different notation for derivatives. It didn't have a conceptual error baked into it (pretending that derivatives just are quotients), but it turns out that that's a useful conceptual error and that's part of why everyone uses Leibniz's notation these days.

If you have an equation of the form dy/dx = f(x) and you want "solve" it, what you are typically looking for is to write y = g(x), right? In other words, the solution to this differential equation is some curve in the x-y plane. This applies more generally, e.g. to situations where you end up with an "implicit" solution like h(y) = g(x): you still get an equation relating x and y which can then be represented as some set of points in the x-y plane (the ones that satisfy that equation).

Now say f(x) happens to have the form a(x,y)/b(x,y). You can consider the system of two differential equations: dy/dt = a(x,y), dx/dt = b(x,y). Solving this system gives x and y as functions of t. Picking any particular value of t gives values of x and y, which gives you a point in the x-y plane. The first key point is that the set of points produced by this procedure as you plug in all possible values of t is exactly the set of points for which the h(y) = g(x) equation above holds. In other words, the solution to the two-equation system encapsulates all the information about the solution to the original equation.

The second key point is that the solution to the two-equation system has _more_ information than the solution to the original equation. In particular, it has the actual values of dx/dt and dy/dt for every given value of t, which don't correspond to anything in our original problem. Their _ratio_ does correspond to something in our original problem: the slope of the tangent line to the solution curve (dy/dx). But the exact values themselves are somewhat arbitrary, as long as their ratio is correct. Put another way, our original problem's solution is a curve in the x-y plane, while the solution of our two-equation system is a curve together with a description for how fast to move along it as t changes. That's the "velocity" bit in Rota's article.

OK, but if how fast we move along the curve doesn't really matter, maybe we can choose to move along it in a nice way that makes it particularly simple to figure out what the shape of the curve is. Our only constraint is that at any given point along the curve the ratio of dx/dt and dy/dt is fixed, because in our original problem we have a fixed dy/dx if we're given values of x and y. So if, at every point (x,y) we multiply dx/dt and dy/dt by the same number (which can depend on x and y) then we get a system of two equations that has different solutions for x and y as functions of t, but the graph of the resulting thing in the x-y plane still looks the same. That's the integrating factor bit; we just formalize it by saying that we multiply both dx/dt and dy/dt by the same function q(x,t), which is exactly what it means to multiply them both at every point by some number that might depend on that point.

The hard part, of course, is choosing a q(x,y) that makes things work out nicely and makes it easy to solve our two equations to get x(t) and y(t).

Here's a concrete example that might help:

Say dy/dx = x/y. We rewrite this in the form dy/dt = x, dx/dt = y. This isn't terribly convenient to solve, so we multiply by q(x,y) = 1/(2xy) to get a new system: dy/dt = 1/(2y), dx/dt = 1/(2x). At this point, maybe you just look at it and go, ah, y = sqrt(t + C1), x = sqrt(t + C2), or maybe you figure out some other way to get there. In any case, now you see that t + C1 = y^2, t + C2 = x^2, so x^2 - y^2 = C for some constant (C2-C1, but both are arbitrary, so this is just some single arbitrary constant). And that's your (implicit) solution for the original differential equation: a hyperbola, or more precisely a family of hyperbolas each of which satisfies the equation.

To illustrate the point about velocities, let's just consider C = 1, so x^2 - y^2 = 1. The point (sqrt(2), 1) lies on this curve. At this point, dy/dx = x/y = sqrt(2). On our original formulation of the parametric system, dy/dt = sqrt(2), dx/dt = 1 at this point. In our reformulation with the integrating factor, dy/dt = 1/2 and dx/dt = 1/(2*sqrt(2)). So the two formulations have us moving along the hyperbola at different speeds at this point as t changes, but they're moving along the same hyperbola.

Does that help at all?

For PDE's it's much more interesting, as the author points out.