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, 06:40 PMdwsmithuu_{xy}-u_{x}u_{y}=0
Find the general solution.

My PDE book doesn't offer explanations so I have no clue on how to approach this problem. - Dec 23rd 2010, 08:36 PMProve It
You probably have to use Separation of Variables.

- Dec 23rd 2010, 08:37 PMdwsmith
How is that done since there is a term u_{xy}?

- Dec 24th 2010, 05:47 PMdwsmith

Should I move this to left and then exponentiate, exponentiate first, or integrated as is? - Dec 24th 2010, 06:18 PMAckbeet
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, 06:19 PMdwsmith
It came from here.

Also, if you differentiate with respect to y, you will obtain this equation.

- Dec 24th 2010, 06:49 PMdwsmith

I think this is how it is done. - Dec 25th 2010, 03:45 AMAckbeet
In looking over your solution, I think your steps are valid. I would point out, however, that your final destination is the same place the usual pde separation of variables got you in post # 5.

Looks good! - Dec 25th 2010, 06:54 AMJester
You could also have achieved the same with the transformation

Just to add, most of the time separation of variables doesn't work for nonlinear PDE's. For example try

- Dec 25th 2010, 01:46 PMAckbeet
Reply to Danny:

Right. Dym's equation was the only nonlinear pde I'd ever come across, before this post, that succumbed to separation of variables. - Dec 25th 2010, 02:35 PMJester
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

?