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.
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.
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.
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.
See: https://en.wikipedia.org/wiki/Phasor
Quoting:
> 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