Differential equation with one univariate function and several variables I have a functional equation where I need to find one univariate function which satisfies conditions involving two variables, for example, find a function $f$ which satisfies, for all $x,y$:
$$
x (x+y) f'(x)(f(y)-f(x)) = (x-y)(f(x)-f^2(x))
$$
How are such equations called? I would like to find literature, but I don't even know where to begin, and what terms to use. This is not an ODE, but also not a PDE.
 A: The direct answer to your question is (as stated in the comments) that the proper term is "functional differential equation".  However, I would caution you about making a full frontal assault on your problem via the methods associated with FDE's.  There are many problems which can be phrased as FDE's, but trying to solve them via general FDE techniques is often very ineffective.
A somewhat general "meta-technique" for solving hard problems involving derivatives$^1$ is to reduce the problem by

*

*Finding a necessary (but perhaps not sufficient) ODE that your solution must solve.

*Determining the solution space of the ODE.

*Looking within the solution space of the ODE for the solution to your original problem.

You can apply this technique to your example problem as follows: Set $y=x+\epsilon$ and take $\epsilon\rightarrow 0$ to find a related ODE $$2x^2 f'(x)^2 = f(x)^2-f(x)$$ which your unknown function $f$ must solve.  This ODE is separable and can be solved analytically!  It is first order, so there is a 1-parameter solution space.  Check through the solution space to determine which solutions satisfy the original equation involving $x$ and $y$, and then you're done.
So you can see how this line of reasoning can substantially constrain your possible solutions.  I would advise trying to make reductions like this before wandering through FDE land too much.

1. For example, the calculus of variations and the method of characteristics can be thought of this way.
