Ten life lessons from differential equations:
- Some problems simply have no solution.
- Some problems have no simple solution.
- Some problems have many solutions.
- Determining that a solution exists may be half the work of finding it.
- Solutions that work well locally may blow up when extended too far.
- Boundary conditions are the hard part.
- Something that starts out as a good solution may become a very bad solution.
- You can fool yourself by constructing a solution where one doesn’t exist.
- Expand your possibilities to find a solution, then reduce them to see how good the solution is.
- You can sometimes do what sounds impossible by reframing your problem.
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10 thoughts on “Life lessons from differential equations”
What about, “When in doubt, try separation of variables?” :)
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.”
Wait, I wrote something potentially meaningful? My keyboard must be broken.
#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.
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.
The 4th order Runga-Kutta method is like a good friend, always helps
I always suspected that Murphy was a statistician.
Related to the above, it’s so much better to fail trying than to not even get into the game, right?
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.
Another one: The solution may depend on the degree of the problem