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[tex]\frac{\partial}{\partial t}N + \frac{1}{\tau}N = -a(t)\frac{\partial}{\partial u}N - u\frac{\partial}{\partial x}N -v\frac{\partial}{\partial y}N + \frac{1}{\tau}D(u,v)\int_{-\infty}^\infty\int_{-\infty}^\infty Ndudv[/tex]

Where

[tex]N = N(u,v,x,y,t)[/tex]

[itex]D(u,v)[/itex] Is a given normalized distribution function. I get to choose this, if you need a particular example, use:

[tex]D(u,v) = \frac{1}{2\pi v_{th}^2}e^{-(u^2+v^2)/2v_{th}^2}[/tex]

[tex]a(t) = A_0\cos(\omega t)[/tex]

[itex]A_0[/itex], [itex]v_{th}[/itex], and [itex]\tau[/itex] are all positive real constants.

And the solution [itex]N[/itex] must be normalizable over [itex]u[/itex] and [itex]v[/itex]