# What is the correct canonical form of this elliptic PDE?

$u_{xx}+2u_{xy}+(1+x^2)u_{yy}=1$

where $\eta = y-x$ and $\psi = \frac{1}{2}x^2$ such that the characteristic curves are given by $\eta\pm\psi i=\text{constant}$.

I have tried and achieved $u_{\eta\eta}+u_{\psi\psi} = \frac{1}{2\psi}$.

Is this correct?