# Nasty Non-linear Differential Equation

I'm trying to come up with some sort of solution to the following differential equation--I warn you, it is ugly. Any help would be much appreciated.

$$D=(1-z)z^2\partial^2_z-[z^2-2\frac{zw(1-z)}{z-w}]\partial_z+(1-w)w^2\partial^2_w-[w^2-2\frac{zw(1-w)}{w-z}]\partial_w$$

$$D^{*}=\frac{zw}{w-z}[(1-z)\partial_z-(1-w)\partial_w]$$

$G_4, E_4, E^{*}$ are all known functions with various parameters.

The differential equation is: $$DG^{*}+D^{*}G_4=G_4E^{*}+G^{*}E_4$$

Are there any techniques that lend themselves well to this type of equation?

Thanks!

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How come $D$ is not simplified? many terms cancel out! – Maesumi Jan 31 '13 at 4:39
I made an error when typing the equation up, sorry! It is now edited. Now notice the second order derivatives. It's pretty awful. – Dick Jan 31 '13 at 4:50
OK. The nature of your equation seems to change substantially depending on the neighborhood of $z,w$ that is relevant for you. So you may want to see where you are attempting a solution and then look for applicable second order numerical PDE solvers. – Maesumi Jan 31 '13 at 4:56
It is so ugly, Dick. – Babak S. Jan 31 '13 at 6:51