Also, what they usually don't tell you is that the recipe bag only works for toy problems. For real applications you most often need numerical approximations.
You can do quite a lot simplifying to get equations that are solvable. A lot of engineering techniques do this, using simplifications that capture the essential behavior of a system (in fact I'd argue that knowing when such approximations are appropriate is one of the core skills of an engineer).
That said, even if you need a numerical solution it will still often require a lot of simplifications in order to be tractable. Multiphase fluid flow, for instance, relies on tons of physics simplifications and empirical correlations in order to make numerical techniques viable.
Or you learn how to make the toy problem that has the same behavior as what you are trying to model. If you don't understand how to simplify problems, you end up just chucking the computation at it and calling it a day. If you have a solution to the basic problem you can give yourself a warm start.
One of my professor used to say that you can do derivatives of every functions, it's mechanical, but you will only be able to find exact solution of an infinitesimal fraction of diffeq, and thinking about that, the same apply with integrals.
Linear circuits are mostly analyzed using the Laplace transform, i.e. in the s-domain, where the differential equations are abstracted away. In the time-domain, simulators are still used most often. But yes a really simple circuit like RC/RL is usually done on the back on the envelope, but then you're talking really simple.
The problem with the analytical approach to differential equations is that it doesn't scale well, and you don't know beforehand whether the approach will work, so you might as well use the numerical approach from the start.
Laplace transformations are differential equations so I fail to see your point. They're just in a different domain. However I do see your point with numerical methods since most complex problems are simulated anyways through simulating software. So in essence, the application becomes pointless because its at such a higher level of abstraction that you don't even have to think about it. You just punch in some numbers and hit analyze and the computer does it all for you.
> Laplace transformations are differential equations so I fail to see your point.
What I mean is that typically an electrical engineer will convert L and C elements to complex impedances (which depend on the frequency through s), and will then compute as though the elements are ordinary resistances. The expression "d/dt" isn't used in the entire analysis.
> the phasor transform thus allows the analysis (calculation) of the AC steady state of RLC circuits by solving simple algebraic equations (albeit with complex coefficients) in the phasor domain instead of solving differential equations (with real coefficients) in the time domain
This is like saying that if I convert mph to m/s then it's not a speed anymore. It's still a differential equation, just in a different domain because you can convert back from the s-domain into the time one.
