[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).