[gus_massa]

> The methods were entirely disconnected.

This is standard an unavoidable. There are like a dozen of tricks that solve a few special cases, and they were found after heroic brute force search in the void. (The real fact that is somewhat hidden is that most differential equations can't be solved analytically. You solve analytically only the few cases that are solvable analytically, otherwise you just get a numerical solution or an approximation.)

> In this way there was not a lot to discuss, and from time to time the professor would gently steer us back to merely copying and memorizing the methods, even though no one had questioned him out loud.

That's a horrible way to teach.

[runnerup]

I phrase it as “The class I cheated my way to an A but didn’t commit any academic dishonesty violation.”

> The real fact that is somewhat hidden is that most differential equations can't be solved analytically. You solve analytically only the few cases that are solvable analytically

I discovered this very early in my semester of Differential Equations. We were allowed a single 8.5x11 notesheet for the exams. As there were only a handful of the “most general” cases which are solvable on paper with a basic calculator, I simply copied the step by step solution for each of the very most general case completely worked out in whatever techniques we were going to be tested on for that exam.

The professor was an engineer before becoming a math professor so he only liked to include real-world situation ODE’s on exams which further reduced the potential problem space.

While it greatly confused the professor/grader who scored my exam that I kept adding zero-coefficient terms before solving the differential equation perfectly…I got 100% on all the exams.

The catch was that I didn’t learn anything. The next semester it turned out that I needed to know those techniques for Reaction Kinetics and Heat&Mass Transfer and Biochemical Engineering (these courses involved deriving and solving many equations from first principles).

I had to crawl back to my Differential Equations professors office hours for 3 weeks and beg him to actually teach me differential equations. He was very confused after asking me what grade I got (an A) and I had to explain to him how I got an A without learning anything.

To his credit, he did a fantastic job assigning me custom work for 3 weeks and reviewing it with me and I was able to learn what I needed for the more advanced courses.

But without his help and some additional tutelage from my peers, I would have been completely screwed for the rest of my Chemical Engineering major.

[aj7]

Well put. Me too.

[eslaught]

> most differential equations can't be solved analytically.

Exactly, so why don't they teach the numerical analysis for actually solving PDEs that matter? These are equations that are very highly relevant to a wide array of real-world science and would be extremely beneficial for many people to know, even if (like calculus or even algebra) most people may not need them later.

I ended up wandering into a career where I work with PDEs nearly every day in some form or other, and would have greatly appreciated some basic training as part of my formal education.

[gus_massa]

Luckily there are many interesting examples that can be solved. In particular the linear differential equations. Many equations can be approximated by a linear version of it.

Also, in Physics, a lot of ODE are mysteriously integrable if the variable is x instead of t. (One reason is that it's easy to measure the force/fields, but the "real" thing are the potential, so you are measuring the derivative of a hopefully nice object.)

Also a lot of the theoretical advanced stuff to prove analytical solutions and to estimate the error in the numerical integrations use the kind of stuff you learn solving the easy examples analytically.

And also historical reasons. We have less than 100 years of easy numerical integrations, and the math curriculum advance slowly. Anyway, I've seen a reduction in the coverage of the most weird stuff like the substitution θ=atan(x/2) (or something like that, I always forget the details). It's very useful for some integrals with too many sin and cos, but it's not very insightful, so it's good to offload it to Wolfram Alpha.

[bee_rider]

Hmm, maybe. How would that impact the larger curriculum? Are you thinking a new class, or just change how differential equations is taught?

I think there is a little bit of an annoying situation where at least Electrical Engineering students are going to want Differential Equations pretty early on as they are pretty important to circuits (IIRC, I don't touch analog stuff anymore). Like maybe as a first semester 200 level class. This doesn't afford space to put a Linear Algebra class in beforehand (needed for numerical analysis).

Maybe the symbolic differential equations stuff could be stuck at the end of integral calculus, but

1) curriculum near the end of the semester is risky (students are feeling done, and it can suffer from schedule shifts).

2) Transfer students or students who satisfied their calc requirements in highschool (pretty common for engineering students) wouldn't be aware of your curriculum changes.

Or, a numerical-focused PDE class could be added to elsewhere. I bet most math departments have one nowadays, but as an elective.

[nautilius]

They do, but if you want to solve PDE numerically instead of DE analytically, you should enroll in the “Numerical Methods for PDE” course instead of “Analaytical methods for DE”.

[physicsguy]

They do, but they’re fairly advanced level courses. For e.g. if you go down the Theoretical Physics or Applied Maths routes you’ll do perturbation theory and asymptotic analysis, probably in Master’s level or grad school.

Most people will do some computational courses that at least have them solving basic PDEs in their first or second year of undergrad now.

(This reflects the state of those in the UK at least)

[plafl]

I had (aerospace engineer) two semesters of Numerical Methods. Then we had more specialized things like Finite Elements Methods.

I wish we had been taught how to use a Computer Algebra System (example, Mathematica or Maple).

[mananaysiempre]

“Unavoidable” is a bit too strong I think. For an ODE course, what does the usual list of elementary methods really include?

- Separation of variables. If one is fine with differentials (or their modern cousins differential forms), there isn’t much to explain here.

- Linear equations solved with quasipolynomials. The only ODE-specific observation is that d/dx in the ( x^k e^x / k! ) basis is a Jordan block; the rest is the theory of the Jordan normal form, which makes interesting mathematical points (an embryonic form of representation theory) but exists entirely within linear algebra (even if it was motivated by linear ODEs historically).

- Ricatti equations. Were always a mystery to me, but it appears they could also be called “projective ODEs” to go with linear ones and have pretty nice geometry behind them (even if, as you said, they were first discovered by brute force search).

- Variation of parameters. Despite the mysterious appearance, this is simply the ODE case of Green’s method beloved in its PDE version by physicists and engineers. (This isn’t often included in textbooks, in fear of scaring students with Dirac’s delta, but Arnold does explain it, and IIRC Courant–Hilbert mentions it in passing as well.)

- Integrating factors. Okay, I can’t really explain what that one means, even though it feels like I should be able to.

Not that teaching it like this would make for a good course (too general, and ODEs ≠ methods for solving ODEs), but that’s essentially it, right? There are certainly other methods you could mention, and not unimportant ones (perturbation theory!.. -ries?), but this basically covers the standard litany as far as I can see. And it’s no haphazard collection of tricks—none of these is just pulling solutions out of a hat.

(In the interest of changing things up and not spending an hour on a single comment, I will omit the barrage of references I’d usually want to include with this list, but I can dig them up if somebody actually wants them.)

[gus_massa]

I was thinking in something similar. (I have no idea what a Ricatti equation is https://en.wikipedia.org/wiki/Riccati_equation ) Perhspas I should have said "half a dozen".

Here the first ODE course is half a semester. If you spend a week or two proving existence and unicity, you get one week to study each method and make a few examples and then you must change to next week trick.

Fourier/Laplace and other advanced stuff are in a more advanced course.

I never used perturbation theory for ODE. I've seen it for solving eigenvalues/eigenvector of operators in QN. But perhaps it's one tool I don't know.