6.103. $\left\{\begin{array}{l}\sqrt{\frac{y}{x}}-2 \sqrt{\frac{x}{y}}=1, \\ \sqrt{5 x+y}+\sqrt{5 x-y}=4 .\end{array}\right.$

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6.103. $\left\{\begin{array}{l}\sqrt{\frac{y}{x}}-2 \sqrt{\frac{x}{y}}=1, \\ \sqrt{5 x+y}+\sqrt{5 x-y}=4 .\end{array}\right.$
Solution.

Given: $\left\{\begin{array}{l}5 x+y \geq 0, \\ 5 x-y \geq 0, \\ \frac{y}{x}>0 .\end{array}\right.$

Let $\sqrt{\frac{y}{x}}=z>0$. Then $z-\frac{2}{z}=1$ or $z^{2}-z-2=0$, where $z \neq 0$. From this, $z_{1}=-1, z_{2}=2 ; z_{1}=-1<0$ is not suitable.

Then $\sqrt{\frac{y}{x}}=2, \frac{y}{x}=4, y=4 x$. From the second equation of the system, we have $\sqrt{5 x+4 x}+\sqrt{5 x-4 x}=4, \sqrt{9 x}+\sqrt{x}=4,4 \sqrt{x}=4, \sqrt{x}=1$, from which we find $x=1$. Then $y=4$. By direct verification, we confirm that this is a solution.

Answer: $(1 ; 4)$.