My question is as follows: What methods can be used to find the set of functions $f:\mathbb{R}\to\mathbb{R}$ satisfying a certain functional equation. An example of a case where this applies is the following:
Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the following equation: $f(x^{3})+f(y^{3})=(x+y)(f(x^{2})+f(y^{2})-f(xy)):\forall x, y\in\mathbb{R}$
I'm curious as to whether there are general methods (or strategies) for solving this type of question, or whether questions like these should just be handled on a case-by-case basis.
Thanks in advance.