Let $\theta$ be an arbitrary constant and $u$ a function of $x,y$.
If I'm given the PDE:
$$ \cos(\theta)\partial^2_xu + 2\sin(\theta)\partial_x\partial_yu -\cos(\theta)\partial^2_yu = 0 $$
How can I reduce it to canonical form?
First I found the discriminant of the PDE and it depends on $\theta$. So the PDE is hyperbolic, elliptic or parabolic on different intervals of $\theta$.
And an extra constraint is to find a solution satisfying:
$$ u\big(\cos(\tfrac{\theta}{2})s,\sin(\tfrac{\theta}{2})s\big) = \cosh(s) = u\big({-\sin}(\tfrac{\theta}{2})s,\cos(\tfrac{\theta}{2})s\big) $$
Where $s$ would be the final variable after having reduced to canonical form.
The problem seems extremely problematic.
My attempt at the solution:
I've tried to get 3 different canonical forms (one for hyperbolic, one for elliptic and one for parabolic) which was pretty lengthy, but I faced problems trying to impose the conditions.
And the notes have left it as an exercise so I can't really check if I'm on the right track.
Is there a theorem or property of such equations that might simplify that I'm missing?
Any help or pointers are appreciated.