Spring 2017, problem 38
Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f : \mathbb{R}^+\to\mathbb{R}^+$ that satisfy $$ \left(1+y~f(x)\right)\left(1-y~f(x+y)\right)=1$$ for all $x,y\in\mathbb{R}^+$.
Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f : \mathbb{R}^+\to\mathbb{R}^+$ that satisfy $$ \left(1+y~f(x)\right)\left(1-y~f(x+y)\right)=1$$ for all $x,y\in\mathbb{R}^+$.