I am trying to solve functional equations in Mathematica, but got nowhere. Just a simple example: $$f(x+y)^2=f(x)^2+f(y)^2$$ for all $x,y \in R$, assuming f is a real-valued function.
How should I approach this problem in Mathematica ?
At first we should try to envision what kind of function it might be. We need no computer system for that. So let's start with assumptons: $f$ is a real valued function on the real domain. Therefore we have: $$ f(x+0)^2=f(x)^2+f(0)^2$$ thus $$ f(0)=0$$ Then $$f(x-x)^2=f(x)^2+f(-x)^2$$ it implies $$0=f(x)^2+f(-x)^2$$ Sum of two non-negative numbers is zero ($f$ is real valued function), therefore for all $x$ $$f(x)=0$$
We need no Mathematica functionality for this conclusion.
Assuming that we need certain functions of the system the problem is more involved and can be analyzed on a case by case basis. At first, basically it depends on the functional relations to be solved. Then we should choose appropriately the class of functions where we can search for solutions.
Assuming, that we are looking for polynomial functions there is e.g.
and among this class the following function yields all solutions (with multiplicity counting):
for third order polynomials:
while for e.g. seventh order polynomials:
We might find other ways but this one demonstrates clearly what kind of problems we encounter.