I would like to know whether this P.D.E is solvable and if yes, for which values of $a$.
$$ 2 (y \cos a + x \sin a) - 4 (y \sin a - x \cos a) f - 2 (y \cos a + x \sin a) f^2 + (x^2 \cos a - 2 x y \sin a - y^2 \cos a) \frac{\partial f}{\partial x} - (x^2 \sin a + 2 x y \cos a - y^2 \sin a) \frac{\partial f}{\partial y} = 0 $$
Where $f = f(x, y)$. I have tried to solve it for the case where $a=0$,
$$ 2 y + 4 x f - 2 y f^2 + (x^2-y^2) \frac{\partial f}{\partial x}- 2 x y \frac{\partial f}{\partial y}=0 $$
on MATHEMATICA but after 3 hours it gave me no result.
Then I utilised the method of characteristics and expressed the $a=0$ case as a system of three ODES
$$ \frac{dx}{dt} = (x^2 - y^2) \\ \frac{dy}{dt} = (-2xy) \\ \frac{df}{dt} = 2yf^2 - 2y - 4xf $$ but MATHEMATICA still gave me no results. So I suspect that this PDE might not have an analytic solution. Is this PDE solvable, and if not could you please indicate a reference which indicates which cases of PDES are non solvable? I am only interested in Analytic solutions.
Any kind of help will be most appreciated. Thank you in advance.
