It is definitely useful to be able to do linear differential equations by hand. It's not useful to keep doing these things by hand. Just like it's useful to kno

It is definitely useful to be able to do linear differential equations by hand. It's not useful to keep doing these things by hand. Just like it's useful to know the algorithm for multiplying two numbers, and it's not useful to keep doing multiplication by hand. What lets people invent new circuits is their good intuition about circuits, not their ability to solve linear differential equations quickly by hand or to compute integrals by hand. When an expert is analyzing a circuit on paper he is thinking about "what happens if the input to this circuit is a sine wave with high frequency", he's not going to solve the differential equations by hand. Rather than circuits look at how electromagnetism or quantum mechanics is taught in college. In my case it was integrals, integrals, integrals. Doing these by hand provided approximately zero intuition into the physics. We could have covered more ground if the instructor would just type these into maple, instead of doing them on the board or in the book by hand. Or how many times have I not had to compute eigenvalues of 2x2 or 3x3 matrices. How many times have we not applied crude approximations in class because doing it by hand was too difficult, when typing it into a compute would give you 100 digits of precision in a couple of milliseconds. One time one of my maths teachers how to compute tan(2) or something like that by hand. After half an hour of calculation he had 2 digits. Computing the integral of something to a crude approximation in an edge case strikes me as futile as computing tan(2) by hand.