GeorgeRP wrote: Find all functions \( f: \mathbb{R}^{+} \to \mathbb{R}^{+} \) such that: \[ f(x^2 + y) = xf(x) + \frac{f(y^2)}{y} \] for any positive real numbers \( x \) and \( y \). Let $P(x,y)$ be the assertion $f(x^2+y)=xf(x)+\frac{f(y^2)}y$ Let $c=f(1)$ Subtracting $P(\sqrt x,1)$ and $P(1,y)$ from $P(\sqrt x,y)$, we get $f(x+y)=f(x+1)+f(y+1)-2c$ Setting there $x=y$, we get $f(x+1)=\frac{f(2x)}2+c$ And plugging this just above : $f(x+y)=\frac{f(2x)+f(2y)}2$ Or also $f(\frac{x+y}2)=\frac{f(x)+f(y)}2$ This is a classical equation whose solution, when lowerbounded (as here) is $f(x)=ax+b$ for some $a,b$ Plugging this back in original equation, we get any $a>0$ and $b=0$ And so $\boxed{f(x)=ax\quad\forall x>0}$, which indeed fits, whatever is $a>0$