# A functional equation problem from Functional Equations (pdf)

Let $\mathbb{R}^+$ denote the set of the positive real numbers. Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ satisfying $$\dfrac{f(f(y)^2+xy)}{f(y)}=f(x)+y$$ for all $x,y \in\mathbb{R}^+$.

I tried to set $x=y=1, x=y=2, x=1, y=2$, so on, but this problem is more difficult.

• Where is this problem from? – the_fox May 11 '18 at 14:55

Put $y=0$. Then the equation shows that $f$ has to be constant. Say $f(x)=c$ for all $x$. Using this in the equation results in $1=c+y$ for all $y>0$ which is impossible. Thus the equation has no solution.

EDIT: As pointed out below the function is defined on $\mathbb{R}_+$. Thus the arguments are not valid.

• $0 \notin \mathbb R^+$ – Fred May 11 '18 at 7:40
• I will edit my answer. – Jens Schwaiger May 11 '18 at 13:57
• $f(x)=x$ is an example, Jens. – Piquito May 11 '18 at 15:07

Proving with $f(x)=kx^n$ we have$$f(k^2y^{2n}+xy)=k(k^2y^{2n}+xy))^n\\f(y)(f(x)+y)=k(f(y)f(x)+yf(y))=k(k^2y^nx^n+ky^{n+1})$$ which is valid only for $k=n=1$

We have an example with $$f(x)=x$$ I hope to find other examples or prove that there are not

• So in the class of monomials there are very few solutions. Nice result. – Jens Schwaiger May 11 '18 at 18:14