[djaque]

I think this gets at the discussion of whether education should provide you with a broad base of knowledge to appreciate a subject as well as use it or whether it should just, in a sense, add to your toolbox the minimum required methods to go into the workforce or move onto more advanced classes.

IMO, I really enjoyed learning about differential equations (DEs) and found that solving them analytically, besides being a nice intellectual challenge, also helped me with advanced courses later on. All of AC circuit analysis and the signals theory classes you take as an electrical engineer, for example, are firmly rooted in the language of DEs and just being able to plug some equations into a computer misses out on all of the real understanding you need to have in those fields to use them in practice.

One of the big red flags a junior electrical engineer can show, in fact, is too much reliance on simulation tools without having a good background understanding of what to expect from them.

[dan-robertson]

Maybe solving some DEs analytically can help one build up intuition as to how the solution may behave, and some algebraic transformations may be useful, but maybe if the goal is for people to know if a solution makes sense then we should put more of an emphasis on the techniques to learn qualitatively about the solution (eg phase diagrams with isoclines and equilibrium points, perturbation analysis, various linearisations).

I wonder how important it is to focus on linearity. On the one hand, lots of equations aren’t linear but on the other hand, most equations are locally linear and knowing about linearity helps with Sturm-Liouville theory.

[neutronicus]

Perturbation theory and linearization all end with you needing to crank a solution to a memorized DE