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Is there a comprehensive set of differential equations on which the method of separation of variables works/doesn't work?

If there is, what makes those differential equations fall into that bracket? Can we get an intuition about the applicability of the method by looking at the physical context?

AkaiShuichi
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  • I once read a chunk of a book that was over 300 pages long and was just proving separation of variables holds in the certain classes of PDEs (all of which were just generalizations of PDEs from physics). There is no reason in general to suspect separation of variables will work mathematically; it often works physically, but if you want to ask that I recommend going to Physics.SE – Brevan Ellefsen Oct 18 '19 at 05:09
  • @BrevanEllefsen Do you remember the name of the book? – Matthew Cassell Oct 18 '19 at 05:26
  • @mattos Just looked. I was thinking of Miller's 1977 Treatise. I actually underestimated the length; it is around 400 pages. I stumbled across it from an MSE post, of which this post is a weak duplicate – Brevan Ellefsen Oct 18 '19 at 05:30
  • @BrevanEllefsen Thanks for letting me know and thanks for the links, I'm going to take a look at it now. – Matthew Cassell Oct 18 '19 at 05:34
  • If you just concentrate on the Laplace equation, the number of coordinate systems where you can separate variables is already quite limited. For example http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.172.7379&rep=rep1&type=pdf – Disintegrating By Parts Oct 20 '19 at 03:55

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