1. ## Find f(x)

If $\frac{x}{y}f\left(\frac{x}{y}\right)=xf(x)-yf(y)$ Find $f(x)$

2. that's easy:

$f(x) = \frac{{\ln x}}
{x}
$

3. And how did you get it??

4. let's define a new function like so:

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

thus we get:

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

does it ring any bells?

5. 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 $f(x)=\frac{(k\ln x)}{x}$