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

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 $\eta=c_2=5y+(1-3i)x$ ?

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  • $\begingroup$ Complex characteristics are no good. I don't have the reference on my, but I'd refer to Fritz John's book, at the beginning of chapter 2 he talks about how to deal with transforming 2nd order PDEs to their canonical forms. Its usually quite difficult for elliptic equations, unlike hyperbolic. $\endgroup$ – Merkh Aug 23 '17 at 20:51
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After change variables

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

we get elliptic equation

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

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