Generally speaking there will be lower order terms. Sometimes (if you're lucky) you can integrate your transformed PDE once or even twice.
I am trying to see a pattern when solving 2nd order linear PDE's.
The canonical form for the Hyperbolic case is
If the discriminant is a real number does the term vanish and hence if the discriminant is some , do we always have some terms?
And similarly for the elliptical case?