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# A weird equation

This is a really hard functional equation by me:

Find all $$f: \mathbb{R} \rightarrow \mathbb{R}$$ such that

$$f(x^2+f(y)^2)=xf(x)+yf(y)$$ for all $$x,y \in \mathbb{R}$$

It looks quite simple but it's very difficult. Good luck if you try it!

Note by Wen Z
7 months, 1 week ago

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simple observations reveals f(x)=x · 6 months, 4 weeks ago

Find all means Find all and prove that you have all of them · 4 months, 3 weeks ago

Comment deleted 6 months ago

yep sure but f(x)=x is the most appropriate one :P · 6 months, 4 weeks ago

Some preliminary observations that might be useful. Will post more things as I keep progressing through.

Switch $$x$$ and $$y$$ around to get

$f(x^2+f(y)^2)=x f(x)+y f(y) = f(y^2 + f(x)^2)$

Substitute $$y=0$$ to get

$f(x^2+f(0)^2)=f(f(x)^2)$ · 7 months ago