If $\displaystyle \frac{x}{y}f\left(\frac{x}{y}\right)=xf(x)-yf(y)$ Find $\displaystyle f(x)$
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that's easy: $\displaystyle f(x) = \frac{{\ln x}} {x} $
And how did you get it??
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?
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 $\displaystyle f(x)=\frac{(k\ln x)}{x}$