ap.analysis of pdes – Beltrami equation

The second order linear PDE

$a(x,y)u_{xx}+ 2b(x,y)u_{xy}+c(x,y)u_{yy}= 0$

was transformed by introducing new independent variable $ξ=φ(x,y)$ and $η=ψ(x,y)$, transforming the PDE to the form

$A(ξ,η)u_{ξξ}+ 2B(ξ,η)u_{ξη}+C(ξ,η)u_{ηη}= 0$


$A=aphi_x^2 + 2bphi_xphi_y+cphi_y^2$



Assume we are in the elliptic case.
Show that in this case we can obtain the situation $A=C$ and $B=0$ in case that $ψ$ and $φ$ are solutions of


and $ψ$ is a solution of the Beltrami equation:

$left(frac{aψ_x+bφ_y}{sqrt{ac−b^2}}right)_x+ left(frac{bψ_x+cφ_y}{sqrt{ac−b^2}}right)_y= 0$.

I’ve tried to calculate $A-C=0$ and $B=0$ and set them even. But I’m stuck at

$frac{phi_x+ipsi_x}{phi_y+ipsi_y} = frac{-bpmsqrt{b^2-ac}}{a}$.

Can you guys please help me? 🙂