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Thread: Find f(x)

  1. #1
    Member great_math's Avatar
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    Find f(x)

    If $\displaystyle \frac{x}{y}f\left(\frac{x}{y}\right)=xf(x)-yf(y)$ Find $\displaystyle f(x)$
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  2. #2
    Senior Member Peritus's Avatar
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    that's easy:


    $\displaystyle f(x) = \frac{{\ln x}}
    {x}
    $
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  3. #3
    Member great_math's Avatar
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    And how did you get it??
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  4. #4
    Senior Member Peritus's Avatar
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    let's define a new function like so:

    $\displaystyle
    w(x)\mathop = \limits^\Delta xf(x)
    $

    thus we get:

    $\displaystyle w\left( {\frac{x}
    {y}} \right) = w(x) - w(y)$

    does it ring any bells?
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  5. #5
    Member great_math's Avatar
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    OK thank you. This is my solution:

    I first differentiated w.r.t x and then w.r.t y and after elimination substitution

    i got $\displaystyle (x^2)(f'(x))+xf'(x)=k$ where k is a constant and after integrating i got $\displaystyle f(x)=\frac{(k\ln x)}{x}$
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