Can anyone determine whether or not this PDE for $f(x,y)$ has a solution? Has anyone encountered a similar PDE before?
$$\alpha x^2 f_{xx} + \beta x f_x - \gamma f - \delta y f_y + \frac{1}{\eta} \left( f f_y - \kappa f_y \right) + \left(1 + w \right) x^{v} y^{w} = 0$$
Where $\alpha, \beta, \gamma, \delta, \eta, \kappa, v$, and $w$ are constants. It is the term $\frac{1}{\eta} \left( f f_y - \kappa f_y \right) $ which is giving me trouble. If this term were absent, this would have a fairly straightforward solution of the form $f = \lambda x^{v} y^{w}$, where $$\lambda = \frac{\left(1 + w \right) }{\delta + \gamma - \beta - \alpha}.$$ Does the term $\frac{1}{\eta} \left( f f_y - \kappa f_y \right) $ make this equation unsolvable or is there another way to obtain a solution?
Any help is greatly appreciated.