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I am doing some physics and I end up with this PDE:

[tex] \frac{\partial q(x,y,t)}{\partial t} = -(x^2 + y^2)q(x,y,t) + ax\frac{\partial q(x,y,t)}{\partial y}[/tex]

where q(x,y,t) is the scalar field to determine and a is a parameter. I need to consider two types of initial conditions: q(x,y,t=0) = 1; and q(x,y,t=0) = delta(x,y).

I have found two tentative solutions:

[tex] q(x,y,t) = C\exp \left(-\lambda s + \frac{y}{ax}(x^2 + y^2/3 - \lambda) \right) [/tex]

where lambda is any (?) number. Another solution is

[tex] q(x,y,t) = C\exp \left(-sx^2 + -y^3/3ax\right) [/tex]

They both seem to satisfy the PDE, but I can't make them satisfy the required initial condition (either 1 or delta function). Any ideas or experience with this kind of equations?

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# 1st order PDE, seems easy but still confusing

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