Find the general solution.

My PDE book doesn't offer explanations so I have no clue on how to approach this problem.

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- Dec 23rd 2010, 05:40 PM #1

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- Dec 23rd 2010, 07:36 PM #2

- Dec 23rd 2010, 07:37 PM #3

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- Dec 24th 2010, 04:47 PM #4

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- Dec 24th 2010, 05:18 PM #5
I'm not sure I think your last steps there are valid. In pde's, separation of variables doesn't look like it does in ode's. You would assume that in which case and Finally, Plugging all this into the pde yields

an identity.

I'm not sure exactly what this means. Does it mean that any solution that is a product of a function of and a function of is a solution to the pde? I'm inclined to think so.

- Dec 24th 2010, 05:19 PM #6

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- Dec 24th 2010, 05:49 PM #7

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- Dec 25th 2010, 02:45 AM #8

- Dec 25th 2010, 05:54 AM #9

- Dec 25th 2010, 12:46 PM #10

- Dec 25th 2010, 01:35 PM #11
There are a few other's. I really like the separation of variables of the Sine-Gordon equation as presented in Debnath's book (rather clever). A very good question is - when does a nonlinear PDE (in variables ) admit (functional) separable solutions in the form

or

?