$f(yf(\frac{x}{y})) = \frac{x^4}{f(y)} $

Question

Solve Functional Equation$$f: \mathbb{R}_+ \to \mathbb{R}_+; \; f(yf(\frac{x}{y})) = \frac{x^4}{f(y)} $$

I'm stuck in the beginning. Any hint will be helpful.

Answer 1

Let $f(1)=k$. Put $x=y=1$ and we get $f(k)=\frac{1}{k}$.

Put $y=x$ and we get $f(kx)=\frac{x^4}{f(x)}$, so $f(x)f(kx)=x^4$ (*)

Put $x=ky$ and we get $f(\frac{y}{k})=\frac{k^4y^4}{f(y)}$, so $f(y)f(\frac{y}{k})=k^4y^4$. That holds for all $y$, so it also holds for $y=kz$ and hence $f(z)f(kz)=k^8z^4$ (**)

Comparing (*) and (**), we have $k=1$. Hence $f(x)=x^2$. It is easy to check that satisfies the original equation.

Answer 2

Let $f(x) = x^2.$ We see that

$$f(yf\left(\frac{x}{y}\right)) = f(y\frac{x^2}{y^2}) = f\left(\frac{x^2}{y}\right) = \frac{x^4}{y^2} = \frac{x^4}{f(y)}.$$