Suppose we're given a function, for example a function $f:\mathbb{C}\rightarrow \mathbb{C}$ such that $f(x)=ax+b $ with $a,b \in \mathbb{C} $. I would like to know which functional equations are satisfied by the given function; but not any functional equation will do as I want to impose some conditions on the equations I'm looking for.

For example, let's consider the equation $\left(x-y\right)f(x+y)=xf(x)-yf(y)$, which the function $f(x)=ax+b $ satisfies. One can see that constants do not appear in the equation, and that it generally gives a relation between $f(x)$, $f(y)$ and $f(x+y)$ and nothing more, as $f$ does not appear in any other form (at least generally). I, for instance, would like to know if there are other equations such that the above function satisfies, that it gives a relation between $f(x)$, $f(y)$ and $f(x+y)$ and nothing more, and that it has no constants appearing.

I would like to know if there are general methods to achieve the above (both specifically as in the example and generally), aside from gaining experience in solving functional equations and dealing with each case individually.