It looks like we agree on the core point, but have different lines of work and experience that colour our impressions of the relative frequency and importance of things. I think it's common to go through school and then never apply some fraction of what you learned in the rest of your career, but it will be a different fraction for each person, depending on their career path. So it's hard to go back and say "I spent so much time studying X, which was not useful in the real world".

As a particularly embarrassing example, I actually gave a presentation to first-year students at my alma mater and told them all that "one thing that never turned out to be useful was integration by parts. It's totally useless to learn these days because Maple and Mathematica are way better at solving integrals than I'll ever be". And then, literally the next week, I had to apply integration by parts to write code for integrating certain arbitrary functions that are passed in as an argument.

> Nonlinear differential equations are more common in the real world than linear ones

It really depends how you look at it. I'd say linear equations are far more common in practical problems, but perhaps common to the point that you hardly notice them and only need to pull out your thinking cap when you hit a nonlinear one. But in a world where I'd never learned how to solve linear equations (other than numerically), I'd probably be really frustrated by how often I encounter them.

> Even when you get to the nitty gritty of real world circuits, you'll encounter nonlinearities and have to deal with them.

Yes. Usually at that point, the main "concept" of the circuit and most modules have already been designed, and you need to pull out SPICE to help design some workarounds when you notice that your capacitor's capacitance is also a function of its voltage, say. You can still think of it as being linearish, with modifications. For most problems, that is (not for everything).

> by looking up lots of (empirical) graphs and tables... Fluid mechanics was one of those courses.

If you're talking about, say, the Moody chart, those nondimensional numbers are what allows you to compress a function of fluid density, viscosity, flow rate, pipe diameter, and pipe roughness (5 inputs) down to a readable chart with just two inputs (Reynolds number and roughness). And someone trying to come up with the empirical measurements to generate such a function would have to take proportionately more measurements, not knowing in advance that most of them will collapse to the same line in Reynolds-number-space.

(Side note: if you're more curious on the fluids side, I really recommend this series of videos from the '50s and '60s: https://www.youtube.com/watch?v=plkDAEx80bQ&list=PLfF--3o8i4... )

> A researcher is unlikely to discover a phenomenon today that is governed by a linear equation

I'm really not trying to be argumentative here, but that's simply not true! First of all, engineers are often more interested in modelling and controlling phenomena than discovering them, and frequently they'll find themselves modelling something that nobody's modelled before. But moreso, I know because even I've done it, and I'm not that special. (It was a novel application of the dynamic Timoshenko beam equation). It's often just a matter of noticing the linear equation that is there, or else pretending that it's linear and then noticing that the result works pretty well.

And again, a lot of interesting non-linear equations reduce to linear ones in the small-signal limit, but remain important in that limit -- for example, most of the domain of acoustics and vibrations deals with small stresses, strains, and displacements, such that all but the most exotic materials behave linear-elastically. But this is still a very active field of research and modelling, and you can bet that design is not just a matter of throwing everything into ANSYS and waiting for a result. Even gravitational waves are studied based on a linearized approximation to the famously nonlinear equations of general relativity.