Canonical forms

Claim: We will show that there will always exists a suitable transformation $\xi=\xi(x,y)$ and $\eta = \eta(x,y)$ such that PDE ()

$$au_{xx} + b u_{xy} + c u_{yy} + d u_x + e u_y + fu = g$$

takes one of the following canonical forms after the transformation:

Form of PDE	Type of PDE
$w_{\xi \xi} = \phi(\xi, \eta, w, w_{\xi}, w_{\eta})$ or	Parabolic eq.
$w_{\eta \eta} = \phi(\xi, \eta, w, w_{\xi}, w_{\eta})$
$w_{\xi \eta} = \phi(\xi, \eta, w, w_{\xi}, w_{\eta})$	Hyperbolic eq.
$w_{\xi \xi} + w_{\eta \eta} = \phi(\xi, \eta, w, w_{\xi}, w_{\eta})$	Elliptic eq.