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**MHB**members,

I have the following equation

$xy(z_{xx}-z_{yy})+(x^{2}-y^{2})z_{xy}=yz_{x}-xz_{y}-2(x^{2}-y^{2})$.

When I transform this into the canonical form via $\xi=2xy$ and $\eta=x^{2}-y^{2}$, I obtain

$z_{\xi\eta}+\frac{\eta}{\xi^{2}+\eta^{2}}z_{\xi}=-\frac{\eta}{2(\xi^{2}+\eta^{2})}$.

This is the point I stuck at.

It follows from here that

$z_{\xi}=\frac{f_{1}(\xi)}{\sqrt{\xi^{2}+\eta^{2}}}-\frac{1}{2}$,

which yields

$z=\int^{\xi}\frac{f_{1}(u)}{\sqrt{u^{2}+\eta^{2}}}{\rm d}u+f_{2}(\eta)-\frac{\xi}{2}$.

Therefore, the given equation has the general solution

$z=\int^{2xy}\frac{f_{1}(u)}{\sqrt{u^{2}+(x^{2}-y^{2})^{2}}}{\rm d}u+f_{2}(x^{2}-y^{2})-xy$.

My question is how to modify the part

$\int^{\xi}\frac{f_{1}(u)}{\sqrt{u^{2}+\eta^{2}}}{\rm d}u$ in a better way

and get rid of the integral if possible.

Thank you very much.

**bkarpuz**