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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
12 months ago

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simple observations reveals f(x)=x

Hiroto Kun - 11 months, 2 weeks ago

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Find all means Find all and prove that you have all of them

Wen Z - 9 months, 2 weeks ago

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Comment deleted 11 months ago

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yep sure but f(x)=x is the most appropriate one :P

Hiroto Kun - 11 months, 2 weeks ago

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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

\[f(x^2+f(0)^2)=f(f(x)^2)\]

Sharky Kesa - 11 months, 3 weeks ago

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