While doing some INMO questions, one entry went this way:
Find all function $ f:\Bbb R\to\Bbb R$ such that $f(x+y)f(x-y)=(f(x)+f(y))^2-4x^2f(y)$.
I made an approach similar to this :
On Putting $x=0, y=0$ we get
$$f(0+0)f(0-0)=(f(0)+f(0))^2-4\times0^2f(0)$$ $$f(0)^2=(2\times f(0))^2$$Which gives us $f(0)=0$.
Then, on putting $x=1, y=1$, $$f(1+1)f(1-1)=(f(1)+f(1))^2-4\times1^2f(1)$$ $$f(2)f(0)=(2\times f(1))^2+-4\times f(1)$$ $$4\times f(1)=4\times f(1)^2$$
Which gives us $f(1)=0 $ or $f(1)=1$.
From here, I can't go further. I think that method I m working on is quite right and will take me to the right answer. But the problem is that I can't find that right answer. I shall be thankful if you can provide me a hint or a complete solution. Thanks.
SIDE NOTE: I am using this method for a while (I got this one from then answer of a post). Now I m thinking to switch, If you know some other method to solve such functional equations, please try to give your answer by that method.