TITLE: Find all solutions to the functional equation $f(x+y)-f(y)=\frac{x}{y(x+y)}$
QUESTION [5 upvotes]: Find all solutions to the functional equation
  $f(x+y)-f(y)=\cfrac{x}{y(x+y)}$

I've tried the substitution technique but I didn't really get something useful.
For $y=1$ I have  
$F(x+1)-F(1)=\cfrac{x}{x+1}$
A pattern I've found in this example is that if I iterate $n$ times the function $g(x)=\cfrac{x}{x+1}$  I have that $g^n(x)=\cfrac{x}{nx +1}$ ,which may be a clue about the general behaviour of the function ( ?) .
I am really kinda of clueless ,it seems like the problem is calling some slick way of solving it.
Can you guys give me a hint ?

REPLY [8 votes]: $f(x+y)-f(y)
=\cfrac{x}{y(x+y)}
=\frac1{y}-\frac1{x+y}
$
so
$f(x+y)+\frac1{x+y}
=f(y)+\frac1{y}
$.
Therefore,
$f(x)+\frac1{x}$
is constant,
so
$f(x)
=d-\frac1{x}
$
for some $d$.
Substituting this,
the $d$s cancel out,
so any $d$ works,
and the solution is
$f(x)
=d-\frac1{x}
$
for any $d$.