rota wrote an essay that discussed differential equations and why it's a misnomer. it's just a grab bag of techniques and if your equation looks a certain way then you can solve with the recipe. but many times the thing you want to find only has a numerical solution and no analytical solution. i've heard that when trying to solve real problems most of the time numerical solutions are the only way. what's your take on this?
It's certainly true that in the "real world" (mathematical modelling for physics, mechanics, industrial chemistry, etc.) you're almost never analytically solving anything, both because you rarely encounter solvable cases and because you don't really care about the analytical solution, engineers love to throw away small terms.
Analytical approaches and especially tools like eigenvectors are I think best reserved for reasoning and understanding, like how a child who knows to break 40 × 8 into something like 4 × 8 × 10 has hopefully understood that multiplication is commutative. Eigenvectors was the point at which control theory really started to make sense to me, although even there in practice you often end up throwing points at a pole-zero plot or doing Kalman filters.
I have no particular background in mathematical pedagogy though.
