Find $f(x) $ given that: $f'(x)=\frac{f(x)-x}{f(x)+x}$

Question

I would appreciate if somebody could help me with the following problem:

Find $f(x)$ given that:

$f \colon \mathbb{R^+} \rightarrow \mathbb{R^+}$, $f$ is differentiable function, and $f'(x)=\frac{f(x)-x}{f(x)+x}$

I tried but couldn't get it that way.

Answer 1

$$y'=\frac{y-x}{y+x}\\y'=\frac{\frac{y}{x}-1}{\frac{y}{x}+1}\\$$now use this substitution $A=\frac{y}{x}$ $$y=Ax\\y'=A'x+A\\A'x+A=\frac{A-1}{A+1}$$now solve for A then find y