# $f(x+y)+f(x-y)= 2x^2-2y^2$. How to find all the solutions to the problem?

Currently I'm learning about functions in my AoPS book. One of the problems is to find all possible solutions to the equation $$f(x+y)+f(x-y)=2x^2-2y^2$$ The book currently explains two methods of solving functional equations. One method is isolation but I don't think that'll work for this equation and another method is to substitute values in for $$y$$ which might work.

My approach: I substituted $$y=0$$ and got $$2f(x)=2x^2$$ so i get $$f(x)=x^2$$ but once I plug my results back into the equation I get $$2x^2+2y^2$$ so this solution is wrong. Now I'm stuck because this is the only approach I have for this problem.

What method should I use to approach this problem and is there even a possible solution?

• You didn't realize, but you have found a contradiction. Hence, you have proven that there is no such function $f$. Oct 15, 2021 at 3:57
• Or, take $y=x$ then $f(2x)=-f(0)$ so $f$ must be constant, but a constant function cannot satisfy the equation.
– dxiv
Oct 15, 2021 at 4:25

You have just shown that there is no solution to the functional equation – deriving $$f(x)=x^2$$ from that equation is a proof that if any solution exists it must be $$x^2$$; showing that this one function does not actually satisfy the equation is a proof that there are no such functions.