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I am a bit confused (and entertained) after reading TEN LESSONS I WISH I HAD LEARNED BEFORE I STARTED TEACHING DIFFERENTIAL EQUATIONS by Gian Carlo Rota

For example, he writes

The most preposterous items are found at the beginning, when the text (any text) will list a number of disconnected tricks that are passed off as useful, such as exact equations, integrating factors, homogeneous differential equations, and similarly preposterous techniques. Since it is rare – to put it gently – to find a differential equation of this kind ever occurring in engineering practice, the exercises provided along with these topics are of limited scope: as a matter of fact, the same sets of exercises have been coming down the pike with little change since Euler

He goes on to mock many other so called "tricks" that he describes as jokes. As a student, these tricks are all I have.

Can someone explain what he means. Is it that, in the real world (whatever that is), the equations are never suitable for these bags of tricks, and therefore we should be learning about numerical methods or analyzing phase portraits, or high performance computing?

I must admit, I am struggling in general to understand how people create differential equations out in the wild. How does one transition from bags of tricks to understanding DE's out in the wild so to speak?

COOLBEANS
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  • One can make analogous arguments about integration - esp. nowadays with powerful computer algebra systems using algorithms that go far beyond the common bag of tricks (e.g. Risch's algorithm and generalizations to LODEs e.g. by Michael Singer). Then the argument boils down to whether or not the tricks are useful more generally, e.g. do they exhibit general problem-solving methodologies that might be useful elsewhere? – Bill Dubuque Oct 12 '19 at 15:43
  • Even in algebra there are analogous examples, e.g. do current students know how to manually compute square and cube roots? Should they? If taught as special cases of general successive approximation techniques (Newton iteration, Hensel lifting, etc) then there is pedagogical value. But there is little value if taught as rote rules. Similarly for integration ODEs (e.g. using differential Galois theory). – Bill Dubuque Oct 12 '19 at 15:43
  • Rota was well-known for his polemics. – Angina Seng Oct 12 '19 at 15:46
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    I'd advice you not to mind him too much. Yes, it's true that most equations that occur in practice may not be susceptible to some of these tricks, yet approximations to them may be. Also, for starters, this makes them familiar with DEs -- the only thing is that one should not let them come away with the impression that all differential equations are so integrable. – Allawonder Oct 12 '19 at 17:55
  • +1 for bringing this lovely essay to my attention. – Jair Taylor Oct 12 '19 at 20:50
  • See [https://math.stackexchange.com/q/1218964/115115](How should a DE course be re-written, if Gian-Carlo Rota is correct?) for an earlier discussion on this subject. – Lutz Lehmann Apr 07 '20 at 06:53

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