Find all functions that satisfy the condition $f(xy,\frac{x}{y})=x^{2}-y^{2}$

Question

Find all functions $f: (0, \infty)^2 \rightarrow \mathbb{R}$ such that for every $x,y >0$ the following equation holds $$f(xy,\frac{x}{y})=x^{2}-y^{2}.$$

Can someone give any tips where to start?

Answer 1

HINT

Make the change of variables: \begin{align*} \begin{cases} u = xy\\\\ v = \dfrac{x}{y} \end{cases} \Longleftrightarrow \begin{cases} u = y^{2}v\\\\ x = yv \end{cases} \Longleftrightarrow \begin{cases} y = \sqrt{\dfrac{u}{v}}\\\\ x = \sqrt{uv} \end{cases} \end{align*}

Can you take it from here?