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Math Help - What functions satisfy the equation?

  1. #1
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    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
    Last edited by Nobody1111; September 7th 2009 at 10:37 AM.
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  2. #2
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    Quote Originally Posted by Nobody1111 View Post
    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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