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The following pde came up in a physics problem: $$ (Cy+D)\frac{\partial^2 u}{\partial x^2}-(Ay+B)\frac{\partial u^2}{\partial y^2}-A\frac{\partial u}{\partial y} =f(x,y), $$ A,B,C,D are fixed constants.

I'm not very experienced at solving pde-s, so before I immerse myself into the subject I would like to ask some experts if they know a simple way to solve it, or know about its solution in the literature.

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Perform a Fourier transform on both variables and turn this second order PDE into a first order one. Solve it then perform an inverse Fourier transform.

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