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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.2012-12-06
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    @Jose Garcia, you have not typing an equation. An equation should have equal sign.2012-12-06
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    sorry i forgot to edit it is an eigenvalue probelm.. A,B and C are constants2012-12-06
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    @Jose Garcia, what is the orginal PDE?2012-12-06
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    it is defined in my last edit :)2012-12-06
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    What's the domain, initial and boundary conditions?2012-12-06
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    assume the domain is a rectangle :D or similar.. i am interested in how to split the solutions $ f(x,y)$ into a differential equation for 'x' and another differential equation for 'y'2012-12-07

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