Consider the following partial differential equation (PDE) of $f(x, y)$:
$$\alpha x^{2} f_{xx} + \beta x f_{x} - \gamma f - \delta y f_y + \eta f^{2} + \theta f f_y y + \lambda + \kappa x^v y^w = 0$$
It is the terms $\eta f^{2} + \theta f f_y y + \lambda$ which make this equation difficult to solve. If they were absent, the equation would reduce to the PDE equivalent of the Cauchy-Euler equation and would have a fairly straightforward particular solution of the form $\omega x^v y^w$, where $$\omega= \frac{\kappa}{\left(\delta w + \gamma \right) - \beta v - \alpha v \left(v-1 \right)}.$$
I have not been able to find a solution when the terms $\eta f^{2} + \theta f f_y y + \lambda$ are included. Has anyone encountered anything like this before? Is it likely to have a closed-form solution?
Any help is appreciated.