# Find all function such that $f(x)-f(y) = (x -y)g(\sqrt{xy})$

Find all functions $f, g$ that satisfy the functional equation $$f(x)-f(y)= (x -y)g(\sqrt{xy}) \quad \forall\ x,y>0.$$

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try it yourself – Applied mathematician Nov 8 '12 at 12:24
Try to obtain an equation that only involves $g$. – Alexei Averchenko Nov 8 '12 at 13:35
I do it and i have this equation: (x^2-1)g(x)-(y^2-1)g(y)=(x^2-y^2)g(xy).If F(x)=(x^2-1)g(x) then we have the equation: ((X^2)*(y^2)-1)(F(x)-F(y))=(x^2-y^2)F(xy). I can't find any solution.Can you help me? – lavinia Nov 9 '12 at 9:37

## 1 Answer

Hint: First fix $y=1$, this shows what $f$ has to be, in terms of $g$. Then fix another $y$ as well.

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I do it and i have this equation: (x^2-1)g(x)-(y^2-1)g(y)=(x^2-y^2)g(xy).If F(x)=(x^2-1)g(x) then we have the equation: ((X^2)*(y^2)-1)(F(x)-F(y))=(x^2-y^2)F(xy). I can't find any solution.Can you help me – lavinia Nov 9 '12 at 9:37