> A course taught as a bag of tricks is devoid of educational value. One year later, the students will forget the tricks, most of which are useless anyway. The bag of tricks mentality is, in my opinion, a defeatist mentality...In an elementary course in differential equations, students should learn a few basic concepts that they will remember for the rest of their lives...

I hated the DE cleass I took in college and it was largely because I felt like it was nothing but a bag of tricks. I very distinctly remember one problem that seemed unsolvable until the teacher showed that you had to substitute a "2" with "1/2 + 3/2". And then, to make matters worse, he put the exact same problem on the test. So we were being rewarded, not for really understanding the core basic concepts, but for memorizing the tricks needed to solve specific problems.

I absolutely love mathematics, for me it is the embodiment of pure beauty. Still, I positively, absolutely hated the sophomore course of ODEs. The way it was taught was extremely abstract: here is the equation, this is integration, this is separation, this is your SLP, now go deal with it.

It was totally pointless and life-sucking. It was not until I got to the 3rd year and learned about specific applications in physics (like heat dissipation, strings, and springs), and later in finance (stochastic calculus) and biology (e.g. Lotka-Volterra) when I realized how many wonderful and extremely useful applications they have.

Have this course started with that, things would be completely different.

> One of many mistakes of my youth was writing a textbook in ordinary differential equations. It set me back several years in my career in mathematics. However, it had a redeeming feature: it led me to realize that I had no idea what a differential equation is.

Wow! Good to see that he wrote this. Looking at his book,

Garrett Birkhoff and Gian-Carlo Rota, Ordinary Differential Equations, Ginn and Company, Boston, 1962.

I got the same impression! I couldn't see what the heck they were driving at. Instead, they seemed to flit around with a lot of tiny topics of little or no interest for little or no reason.

Want to understand ordinary differential equations, read Coddington:

Earl A. Coddington, An Introduction to Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, 1961.

Then for more, to make such equations much more important, read some deterministic optimal control theory, e.g., Athans and Falb

Michael Athans and Peter L. Falb, Optimal Control: An Introduction to the Theory and Its Applications, McGraw-Hill Book Company, New York,

that, BTW, also has some good introductory, but very useful, material on ordinary differential equations.

More generally, want to know what to study in a subject that will be useful? Okay, one approach is to go to more advanced material that is an application of that subject and see what that material emphasizes for prerequisites, e.g., sometimes quite clear in an appendix.

E.g., Athans and Falb say quite clearly what is important in ordinary differential equations for their work.

The internet changed drastically the process of writing books. I saw people making profit from books available online for free. I saw books written chapter by chapter with errors found quickly by first readers. I saw systems that allow commenting parts that are not clear enough with comments how to clarify them.

If you promise to deliver a textbook that teaches skills relevant to engineers, they will fund the time it takes to write the book. I would spare couple of dollars even if you said that it will take 5 years. I believe, there will be even companies that will give you funds upfront. If you reach out for help there will be people who will help you collect example problems from different fields to replace couple of "salt tank problems".

I am not a mathematician, so I don't know how mathematics textbooks are written and how much effort goes into them, so feel free to point out that this idea is stupid.

They had a custom textbook created for their 2-course ODE sequence that several of the faculty collaborated on. Though it did contain content on uniqueness theorems and some proofs, far and away the biggest two items hammered in were (a) linear equations with constant coefficients, and (b) Laplace transform methods.

They also offered (at the time) a 3rd, optional course called Boundary Value Problems that was focused on several physics-motivated BVPs like with Laplace's equation, heat equation, wave equation, Young's modulus, and others, and that course heavily used Fourier and Laplace methods.

We did have word problems, but they were almost exclusively "salt tank" problems. Literally, every word problem described a tank of water or pre-mixed brine solution, with some description of either more salt or more water being added or removed, either gradually or in discrete injections.

The fact that every problem was an infamous "salt-tank problem" essentially made its status as a word problem irrelevant. This seems like it wouldn't be that helpful but actually it was really nice. You got so used to the different pieces that comprised the modeling problem that when you went off and did something in other courses, like circuit systems or conservation systems in mechanical engineering, you knew how to translate the problem to 'salt tank' form, which really covered a huge range of practical problems.

As a math major, one fault I noticed of this method was that it did not make the connections to linear algebra very clear. It took me another few semesters afterward to catch up on that part, but I can understand how engineering majors cared less about that.

I don't know what Rose-Hulman does for this curriculum now, but it would be cool to somehow take a "snapshot" of their methods for it and compare it with other experiences like this OP.

I feel like this is impossible without going to the complex plane. Like the author said, taking the inverse Laplace transform is no joke.

I feel like I never properly understood the Laplace transform until I learned about Landau damping. This is when waves exist, but are damped in a collisionless plasma. This damping is not disspiation and the energy does not get converted into heat. The usual way of presenting this is to show that if one Fourier transforms in time, you get the wrong answer. The fact that the system begins at a certain state, and is thus an initial value problem, needs to be respected.

I don't get this. Differential equations theory is about proving existence and uniqueness of solutions. If you have to use numerical techniques to actually compute the solution, then that's perfectly fine. After all, even if the solution is explicit, like sin(x), or especially a special function, then we still need to use numerical techniques to actually evaluate that explicit solution.

I've been collecting interesting scientific papers and publications since early 2000 (I've a collection of 10,000 or so) and I've not yet seen a single academic, not even a computer scientist, who understands how to name your documents right so that when I download them I could quickly find them. I've to rename every single pdf. It's infuriating.

Someone should teach academics an SEO course.

[1] Review here: http://www.maa.org/publications/maa-reviews/indiscrete-thoug...

Mathematicians have so much fun..

In my opinion, the ideal way of learning would be to first have very basic (only conceptual) introductory courses of applied fields, where we find some basic equations that we do not know how to solve. And then, we study DE to learn the techniques to solve these problems, avoiding direct references but keeping in mind where we are going with all this.

I am now lusting after http://www.amazon.ca/Exterior-Analysis-Using-Applications-Di... but it's a bit pricy for a indulgence purchase!

Professional mathematicians have avoided facing up to density functions by a variety of escapes, such as Stieltjes integrals, measures, etc. But the fact is that the current notation for density functions in physics and engineering is provably superior, and we had better face up to it squarely

> It is of the utmost importance to explain the relation between the solutions of the differential equation and the solutions of the system. The solutions of the system are trajectories, they are parametric curves endowed with a velocity given by the vector field. The solutions of the corresponding differential equation are integral curves, and their graphs are the graphs of the trajectories deprived of velocity. Often, instead of solving the differential equation, it is more convenient to solve the corresponding autonomous system.

At ANSTO I worked with solution of simultaneous first order differential equations, as arose from Newtons law of cooling for a 4-body calorimeter. That was fun.

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...)

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.

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.

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?