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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?

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

1 Answer 1

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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.

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