Canonical form of an elliptic PDE

Question

$5 \frac{\partial ^2z(x,y)}{\partial x^2}-2 \frac{\partial ^2z(x,y)}{\partial x \partial y}+2\frac{\partial ^2z(x,y)}{\partial y^2}=0$

Question: Solve the PDE by transforming canonical form.

Solution: Since $\Delta=-36<0$, this is a elliptic PDE and we have complex characteristics such that $c_1=5y+(1+3i)x$ and $c_2=5y+(1-3i)x$.

Then, will we select $\xi$ and $\eta$ such that $\xi=c_1=5y+(1+3i)x$ and $\eta=c_2=5y+(1-3i)x$ ?

Answer 1

After change variables

$$\xi=x+3y,\quad \eta=3x$$

we get elliptic equation

$$45z_{\xi\xi}+45z_{\eta\eta}=0.$$