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I am stuck with the separation of variables for the following PDE:

$$ -Ay^{2}\partial _{y}^{2}f(x,y)-y^{2}\partial _{x}^{2}f(x,y)+iBy \partial _{x} f(x,y)+C= \lambda _{n}f(x,y)$$

Here, $A, B, C$ are constants and $ i = \sqrt{-1} $.

I believe that the solution to the equation of $y$ should be a Bessel function but I don't know how to split this into two linear equations in the variables $x$ and $y$.

To eliminate the dependence on $x$, I could take the Laplace transform so I get a function of $ f(x,s) $, but I do not know what else to do.

Any hints?

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I believe I can't C and equation here. –  vanguard2k Dec 6 '12 at 16:28
    
@Jose Garcia, you have not typing an equation. An equation should have equal sign. –  doraemonpaul Dec 6 '12 at 17:39
    
sorry i forgot to edit it is an eigenvalue probelm.. A,B and C are constants –  Jose Garcia Dec 6 '12 at 18:31
    
@Jose Garcia, what is the orginal PDE? –  doraemonpaul Dec 6 '12 at 18:40
    
it is defined in my last edit :) –  Jose Garcia Dec 6 '12 at 18:42
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