interesting fun eq

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zadid xcalibured
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interesting fun eq

Unread post by zadid xcalibured » Sat Mar 10, 2012 8:24 pm

find all functions f such that for all real x,y \[f:\mathbb{R}\rightarrow \mathbb{R},f(x^3+y^3)=x^2f(x)+yf(y^2)\]

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*Mahi*
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Re: interesting fun eq

Unread post by *Mahi* » Sun Mar 11, 2012 1:19 pm

The hardest part here is to prove $f$ is continuous.
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Sakal
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Re: interesting fun eq

Unread post by Sakal » Mon Mar 12, 2012 3:45 am

f(x)=cx is (c arbitary constant) [also,includes, c=0 means f(x)=0 ]
one solution I got..it's continuous .yet ,no other solution I got . trying.....

Corei13
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Re: interesting fun eq

Unread post by Corei13 » Mon Mar 12, 2012 7:33 pm

Denote the statement by $P(x,y)$.
$P(x,0),P(0,x)$ imply that $f$ is a Cauchy equation and $f(x^2)=xf(x)$
Now, $(x+1)(f(x)+f(1))=(x+1)f(x+1)=f((x+1)^2)=f(x^2 + 2x + 1)$
$=xf(x)+2f(x)+f(1)$
So, $f(x)=xf(1)$.
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