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Several of the common partial differential equations encountered in physics are mentioned as easy to solve using the variable separable method. However I do not understand how does one guarantee that the solutions generated using this method is the complete set of solutions. I am finding it quite hard to believe that such a method can produce the entire set of solutions.

Can anyone explain if it indeed does generate the complete set of solutions to a PDE. If not, a counterexample would work. Also, I would like to know if we use this because it is consistent and easy to work with?

saisanjeev
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  • A lot of it comes down to uniqueness of solutions. If you find a solution by any means, and you can prove from the form of the pde that solutions are unique, then you're good to go. This is not always possible, of course. – Adrian Keister Nov 05 '19 at 16:31
  • See also https://math.stackexchange.com/questions/575205/why-separation-of-variables-works-in-pdes, https://math.stackexchange.com/questions/1163773/when-does-separation-of-variables-yield-basis-of-solution-set – Hans Lundmark Nov 05 '19 at 16:50

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