Example 2 Solve the equation $\sqrt{x-\frac{1}{x}}+\sqrt{1-\frac{1}{x}}=x$.
Let $u=\sqrt{x-\frac{1}{x}}, v=\sqrt{1-\frac{1}{x}}$.
Then $u+v=x \neq 0, u^{2}-v^{2}=x-1$.
Dividing the two equations gives $u-v=1-\frac{1}{x}$.
Thus $2 u=x+1-\frac{1}{x}=u^{2}+1$
$\Rightarrow(u-1)^{2}=0 \Rightarrow u=\sqrt{x-\frac{1}{x}}=1$
$\Rightarrow x=\frac{1+\sqrt{5}}{2}$ (negative value discarded).