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Math Help - Find f(x)

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

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


    f(x) = \frac{{\ln x}}<br />
{x}<br />
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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:

    <br />
w(x)\mathop  = \limits^\Delta  xf(x)<br />

    thus we get:

    w\left( {\frac{x}<br />
{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 (x^2)(f'(x))+xf'(x)=k where k is a constant and after integrating i got f(x)=\frac{(k\ln x)}{x}
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