The most common view of differential equations may be sheer terror, but those who get past terror may have one of the following perspectives.
Naive view: All differential equations can be solved in closed form by applying one of the 23 tricks covered in your text book.
Sophomoric view: Differential equations that come up in practice can almost never be solved in closed form, so it’s not worth trying. Learn numerical techniques and don’t bother with analytic solutions.
Classical view: Some very important differential equations can be solved in closed form, especially if you expand your definition of “closed form” to include a few special functions. Analytic solutions to these equations will tell you things that would be hard to discover from numerical solutions alone.
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I never held the naive view; I learned the sophomoric view before I knew much about differential equations. There’s a lot of truth in the sophomoric view — that’s why it’s called sophomoric. It’s not entirely wrong, it’s just incomplete. (More on that below.)
I’ve learned differential equations in a sort of reverse-chronological order. I learned the modern theory first — existence and uniqueness theorems, numerical techniques, etc. — and only learned the classical theory much later. I studied nonlinear PDEs before knowing much about linear PDEs. This may be the most efficient way to learn, begin with the end in mind and all that. It almost certainly is the fastest way to get out of graduate school. But it’s not very satisfying.
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I get in trouble whenever I mention etymologies. So at the risk of sounding like Gus Portokalos from My Big Fat Greek Wedding, I’ll venture another etymology. I’ve always heard that sophomore comes from the Greek words sophos (wise) and moros (fool), though something I read suggested this may be a folk etymology. It doesn’t matter: regardless of whether that is the correct historical origin of the word, it accurately conveys the sense of the word. The idea is that a sophomore has learned a little knowledge but is over-confident in that knowledge and doesn’t know its boundaries. In mathematical terms, it’s someone who has learned a first-order approximation to the truth and extrapolates that approximation too far.
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