Consider the equation $$u_{xx} +2ay u_{xy} +e^{2x} u_{yy} −u = 0,$$ where $a$ is a real constant. Determine the value(s) of a when this equation is elliptic everywhere in the $xy$-plane in which case find the canonical variables and reduce the equation to its canonical form. Explain where the canonical equation degenerates.
I just can't seem to find the canonical variables for this equation.