How to recover original function from a modified argument

Question

In an auto-evaluation I found this problem:

Find $f(x, y)$ if $f(x + y, y/x) = x^2 − y^2$

The solution given was:

$\frac{x^2(1 − y^2)}{(1 + y)^2}$

During my try I found that $f(x,0)=f(x+0,0/x)=x^2$ but not much more.

How is this mean to be solved? Is there a methodology instead of just trying to guess? Thank you!

Answer 1

HINT

Consider the following change of variables: \begin{align*} \begin{cases} u = x + y\\\\ v = \dfrac{y}{x} \end{cases} \end{align*}

Now solve for $x$ and $y$.

Can you take it from here?

Answer 2

If $$x=u+v$$ $$y=\frac{u}{v}$$ then $$v=\frac{x}{y+1}$$ $$u=\frac{xy}{1+y}$$ Hence $$f(x,y)=f\left( u+v , \frac{u}{v} \right) =u^2 -v^2 =(u-v)(u+v)=x^2 \frac{y-1}{y+1}$$

Answer 3

Let $x+y=u$ and $\frac{y}{x}=v$. Then $x=\frac {u}{1+v}$ and $y=\frac{uv}{1+v}$ ,therefore $x^{2}-y^{2}$=$\frac {u^{2}(1-v^{2})}{(1+v)^{2}}$ and change $u$ to $x$ and $v$ to $y$.