Question 42
Let $\mathbb{R}_{\gt 0}$ denote the set of positive real numbers. Find all functions $f: \mathbb{R}_{\gt 0} \rightarrow \mathbb{R}_{\gt 0}$ such that
$$
x+f(y f(x)+1)=x f(x+y)+y f(y f(x))
$$
for all positive real numbers $x$ and $y$.
$$
x+f(y f(x)+1)=x f(x+y)+y f(y f(x))
$$
for all positive real numbers $x$ and $y$.