$u_{xx} + u_{xy} + u_{yy} = 0$ in canonical form

Question

How do i put $u_{xx} + u_{xy} + u_{yy} = 0$ in canonical form?

$a=1, b=1/2, c=1 $ implies that it is elliptic as $b^2 - ac <0$

$dy/dx = \lambda$ where $a\lambda^2-2b\lambda+c=0$ gives $\lambda = \pm\sqrt{\frac{-3}{4}} + 1/2$

As these roots are complex I am not sure how to proceed?

Can i take $ξ=y-x$ and $η=x$ to give me $u_{ξξ} + u_{ηη} - u_{ηξ} =0$ which is in canonical form?

Answer 1

$\lambda_1=\frac{1+i\sqrt{3}}{2},\ \lambda_2=\overline{\lambda_1}=\frac{1-i\sqrt{3}}{2}\\ \implies \varepsilon=y-\lambda_1x, \ \eta=y-\overline{\lambda_1}x$

The standard procedure to deal with complex roots involves finding the canonical form by making the substitution $\alpha=\frac{\varepsilon+\eta}{2}=y-\frac{\lambda_1+\overline\lambda_1}{2}x=y-Re(\lambda_1)x,\ \beta=\frac{\eta-\varepsilon}{2i}=\frac{\lambda_1-\bar\lambda_1}{2i}x=Im(\lambda_1)x$.

The canonical form so obtained will not contain complex constants.