Question 29
Let $c\gt 0$ be a given positive real and $\mathbb{R}_{\gt 0}$ be the set of all positive reals. Find all functions $f: \mathbb{R}_{\gt 0} \rightarrow \mathbb{R}_{\gt 0}$ such that
$$
f((c+1) x+f(y))=f(x+2 y)+2 c x \quad \text{ for all } x, y \in \mathbb{R}_{\gt 0} .
$$
$$
f((c+1) x+f(y))=f(x+2 y)+2 c x \quad \text{ for all } x, y \in \mathbb{R}_{\gt 0} .
$$