Three views of differential equations

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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15 thoughts on “Three views of differential equations

  1. I must admit that I went through a variant of the naive view, which I will call the Non-mathematics-major View: Almost all of the problems that I see in this class will be solvable by one of the 23 tricks in the textbook. (The exceptions will have been specifically addressed by the professor.)

    Let’s be honest; It’s a cop-out. Like many such approaches, it has repercussions later in life: It later leads to the Pragmatist’s View, which calls for finding someone who majored in maths (the next level of cop-out), or if that doesn’t work out, adopting the Sophomoric View.

    -Ben

  2. There is a fourth view. The Qualitative view. You cannot solve most of the equations, but there are a lot of techniques which yield a very accurate description of the behavior of their solutions. This is very useful when you are using numerical methods. One very powerful result is for instance the central manifold theorem, which gives a method to reduce a large system to a smaller one, with av accurate estimate of the error.

    I like your blog! r

  3. John, are you including, so called, weak solutions in the Classical view. For example, solutions to conservation pdes (Burgers), Hamilton-Jacobi equations, or weak solutions to elliptic pdes using Sobolev spaces? If so, the name is really confusing. Initially, I was thinking the classical view was looking for “classical solutions” to pdes. However, if weak solutions are not included in the classical view, at least one more category is needed.

    The other two things that weren’t included is Dynamical Systems perspective. You learn about a pde by studying the bifurcations, critical points, and stability properties of the pde. Additionally, Asymptotics is also really important when studying pdes, but could probably be put in the “Sophomoric view”.

    Anyways, whenever you create a categorical system, there is always exceptions. However, that misses the point when understanding people’s basic beliefs about pdes.

  4. When I mentioned “classical,” I had in mind the perspective of mathematicians a century ago. They were well aware that arbitrary differential equations couldn’t be solved analytically, but they also knew an awful lot about important equations that could be solved analytically. I didn’t have in mind “classical” as in “classical solutions.”

    By “modern” I have in mind the kinds of things you mention: weak solutions, Sobolev spaces, etc.

    I didn’t mean the three views I listed to be exhaustive. But I’d put asymptotics in with the classical view.

  5. In hindsight, a more accurate (but verbose) title would have been “Three views on analytic solutions to differential equations.” The three views would be that they (1) always exist, (2) never exist, or (3) sometimes exist. Or maybe (1) they’re everything, (2) they’re unimportant, or (2) they’re not everything but they’re sometimes important.

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  7. I was working through a book on nomography. Before computers were common, we used drafting techniques to solve differential equations.

  8. Followed Steven Strogatz excellent dynamical systems you tube videos showing the depth of understanding, without solving, using graphical analytical methods.

  9. I was a chemistry major in college. Please excuse me if what I am about to relate is nothing more than foolish. My idea- as differential equations become more complicated (more variables as opposed to more parameters) they become less stable in application. Less likely to calculate realistic real world results. Is my idea still correct? Was my idea ever correct? Now that we have super powerful computers are equations (unstable in the past) becoming more and more stable? Predicting the weather is my idea of a perfect example of this thesis. Not too difficult to calculate conditions for the following day. Almost impossible to calculate conditions for next week. Anyone care to comment? Can you lead me to a literature source?

  10. More complicated equations are not necessarily less stable. If a linear system has a million variables, but all eigenvalues are negative, then the system is stable. If a system has two variables, with one positive and one negative eigenvalue, the system is unstable. One could argue, however, that with more variables, it’s less likely that nice properties are likely to hold, such as all eigenvalues being negative.

    Nonlinear systems are harder to understand, but there too the qualitative properties of the system are more important than the size of the system.

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