Life lessons from differential equations

Ten life lessons from differential equations:

  1. Some problems simply have no solution.
  2. Some problems have no simple solution.
  3. Some problems have many solutions.
  4. Determining that a solution exists may be half the work of finding it.
  5. Solutions that work well locally may blow up when extended too far.

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10 thoughts on “Life lessons from differential equations

  1. Excellent post.

    On the separation of variables idea by Joseph, something that is remarkable to me is that, even in technical situations in engineering and software teams, it’s amazing how often people seem to want to confound “variables” that began completely separated and, once confounded, make thinking about the system much harder. That moved Dijkstra to write about “A proper separation of concerns.”

  2. #9 may need some explanation. The usual way to show that a PDE has a smooth solution is to first show that the equation has a weak solution in some space of more general functions. This involves #10, working with functions that aren’t necessarily differentiable in the classical sense.

    Then you show that any weak solution to the equation is in fact smooth. It’s analogous to showing an equation has an integer solution by first showing it has a real solution (e.g. by using calculus), then showing that the real solution is in fact an integer.

  3. Sometimes a quick-and-dirty approximate solution now is better than the exact solution later.

    One difference here: sometimes in DEs the approximate solution (from asymptotics, perturbation theory, etc.) yield so much more insight into a problem than a messy exact solution, that the approximate solution now may actually be more helpful than the exact solution now.


  4. Related to the above, it’s so much better to fail trying than to not even get into the game, right?

  5. Another lesson: if there’s a single (attracting, stable) state, that’s where a system will tend to. It might look different for a while, but it will get there.

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