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In the last few weeks of a course I've been taking (mathematical methods for physicists) we've learned about how to solve PDEs using the method of separation of variables. I think I pretty much know how to do it, technically. The one thing which I can't seem to grasp is why the most general solution is the linear combination of all of the separated forms. How do I know that there aren't more solutions which agree with some specific boundary conditions, while the separation method does not?

Thanks

GoingWeb
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  • related: https://math.stackexchange.com/questions/184110/does-solve-pde-by-combination-of-variables-always-cannot-find-the-general-soluti and https://math.stackexchange.com/questions/402545/how-do-we-know-that-pde-solutions-obtained-via-separation-of-variables-are-the-o and https://math.stackexchange.com/questions/863740/when-is-separation-of-variables-an-acceptable-assumption-to-solve-a-pde – Henry Jan 13 '18 at 10:46
  • The comments to the question posted in the second link seem to claim that the "general solution" gained by the separation method is actually not the most general. It makes perfect sense to me, and I would love to get more information on that or references which further agree with that statement. – GoingWeb Jan 13 '18 at 11:03
  • Typically you show that you can solve a given class of boundary and/or initial value problems using separation of variables, and then you establish uniqueness theorems to show that there is only one such solution under suitable conditions. The regions and equations need to be amenable to separation of variables. – Disintegrating By Parts Jan 13 '18 at 23:45

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