When studying a computer vision problem I end up with a function $f(x,t)$ that satisfying $\frac{\partial f}{\partial t}+f\frac{\partial f}{\partial x} =\frac{2f^2}{x}$.
My question includes two parts:

1. What are the solutions to the above equation in general? I worked out three specific solutions: (1) $ f(x,t)= \frac{x}{k-t}$. (2) $ f(x,t)= kx^2$. (3) $ f(x,t)= \frac{kx^2-x}{b+t}$ (k and b are constants). I have no idea whether there is other solutions.
2. To resolve the equation in another point view, we can define an operator $\Omega f=\frac{\partial f}{\partial x} - \frac{2f}{x}$. Suppose we can find the eigenfunctions of the operator $\Omega$, namly $\Omega f=\lambda f $ ($\lambda$ is constant), so that any solution $f$ can be represented by linear sum of these eigenfunctions (very similar to Schrödinger equation). 
Unfortunately, no more functions other than the three above are found, though I expect that there should be some more complicated eigenfunctions, such as Fourier series. Now I am looking into wavelet theories to find a new clue.

Any suggestions on the general solution to $\frac{\partial f}{\partial t}+f\frac{\partial f}{\partial x} =\frac{2f^2}{x}$, or to the eigenfunctions for operator $\Omega f=\frac{\partial f}{\partial x} - \frac{2f}{x}$?