Question 97
Find all functions $f:(0, +\infty) \rightarrow (0, +\infty)$ for which for any positive numbers $x, y$ the following equality is true:
$$
f(f(x)+y) = f(f(x))+2yf(x)-f(y)+2y^2+1. \qquad (\text{Ihor Voronovych})
$$
$$
f(f(x)+y) = f(f(x))+2yf(x)-f(y)+2y^2+1. \qquad (\text{Ihor Voronovych})
$$