I'm learning about the second order PDEs, and I've come across the transformation of variables I don't understand.

We have $\displaystyle u=u(x, y)$ and a, b, c functions of class $\displaystyle C^2$, which are not at the same time all equal to zero.

The equation in question is:

(*)$\displaystyle a \frac{\partial^2 u}{\partial x^2} + 2b \frac{\partial^2 u}{\partial x \partial y} +c \frac{\partial ^2 u}{\partial y^2} + f(x, u, \nabla u)=0$

We introduce the change of variables:

$\displaystyle \xi=x+y$, $\displaystyle \eta=x-y$

How does the equation (*) now looks like?

I've tried googling change of variables, but I'm still lost.

Please, help.

Thank you.