Using the characteristics to get the canonical form of a pde

I'm using subscripts as partials to save time

\begin{align} u_x &= u_\xi\xi_x + u_\eta\eta_x = - u_\xi + u_\eta \\ u_y &= u_\xi\xi_y + u_\eta\eta_y = u_\xi \end{align}

Using the chain rule again

\begin{align} u_{xx} &= (-u_\xi + u_\eta)_x = -u_{\xi\xi}\xi_x - u_{\xi\eta}\eta_x + u_{\eta\xi}\xi_x + u_{\eta\eta}\eta_x = u_{\xi\xi} - 2u_{\xi\eta} + u_{\eta\eta} \\ u_{yy} &= (u_\xi)_y = u_{\xi\xi}\xi_y + u_{\xi\eta}\eta_y = u_{\xi\xi} \\ u_{xy} &= (u_\xi)_x = u_{\xi\xi}\xi_x + u_{\xi\eta}\eta_x = -u_{\xi\xi} + u_{\eta\xi} \end{align}

Combining everything:

$$ 3u_{xx} + 6u_{xy} + 3u_{yy} - u_x - 4u_y + u = 3u_{\eta\eta} - 3u_\xi - u_\eta + u = 0 $$

which is indeed parabolic