What functions satisfy the equation?

**Nobody1111** · September 7th 2009, 09:00 AM

Hello,

The problem I'm struggling with is: Find all continuous functions $f$ such that $f(0)=0$ satisfy the equation:

$af^2(-x)+2f(-x)-f(-2x)=0,$
where $x \in R$ and $a>0$ is a parameter?

Surely, $f(x)=0$ satisfies this equations. But is it the only function?

Thanks for help

**Opalg** · September 7th 2009, 12:56 PM

Originally Posted by Nobody1111

Find all continuous functions $f$ such that $f(0)=0$ satisfy the equation:

$af^2(-x)+2f(-x)-f(-2x)=0,$
where $x \in R$ and $a>0$ is a parameter?

Surely, $f(x)=0$ satisfies this equations. But is it the only function?

(I'm assuming that $f^2$ means f squared rather than the other possible meaning f composed with itself.)

For every real number c, there is a solution $f(x) = a^{-1}\bigl(e^{cx} - 1\bigr)$ . I suspect that these are the only continuous solutions, but I'm not sure of that.

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