I encountered this problem while studying the value of a firm which faces uncertainty over the price of its output good. Consider the following nonlinear ODE for $f(x)$ where all Greek letters are constants and $f_x$ denotes the first derivative of $f(x)$ with respect to $x$:
$$\alpha x^2 f_{xx} + \beta x f_{x} + \gamma f + f^2 + \theta + x^{\eta} = 0$$
If the terms $f^2$ and $\theta$ were absent, this equation would reduce to a Cauchy-Euler equation with the following general solution:
$$f(x) = A_1 x^{\phi_1} + A_2 x^{\phi_2} + \frac{x^{\eta}}{\alpha \eta^2 +\left(\beta - \alpha \right) \eta + \gamma}$$
Where $A_1$ and $A_2$ are constants of integration and $\phi_1$ and $\phi_2$ are the roots of the following quadratic:
$$\alpha \phi^2 + \left(\beta - \alpha \right) \phi + \gamma = 0$$
However, with the terms $f^2$ and $\theta$ included it becomes a second order nonhomogeneous Riccati equation, which I cannot solve. Does anyone know a method of solving this type of differential equation? Has anyone encountered a similar equation before?
Any help is appreciated.