# Math Help - What functions satisfy the equation?

1. ## Which functions satisfy the equation?

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

2. 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.