Simplify: $\;$ $\left(\dfrac{x^{\frac{1}{2}} - y^{\frac{1}{2}}}{x y^{\frac{1}{2}} + y x^{\frac{1}{2}}} + \dfrac{x^{\frac{1}{2}} + y^{\frac{1}{2}}}{x y^{\frac{1}{2}} - y x^{\frac{1}{2}}}\right) \dfrac{x^{\frac{3}{2}} y^{\frac{1}{2}}}{x + y} - \dfrac{2y}{x - y}$
$\left(\dfrac{x^{\frac{1}{2}} - y^{\frac{1}{2}}}{x y^{\frac{1}{2}} + y x^{\frac{1}{2}}} + \dfrac{x^{\frac{1}{2}} + y^{\frac{1}{2}}}{x y^{\frac{1}{2}} - y x^{\frac{1}{2}}}\right) \dfrac{x^{\frac{3}{2}} y^{\frac{1}{2}}}{x + y} - \dfrac{2y}{x - y}$ $\;\;\; \cdots \; (1)$
Consider $\; \;$ $\left(\dfrac{x^{\frac{1}{2}} - y^{\frac{1}{2}}}{x y^{\frac{1}{2}} + y x^{\frac{1}{2}}} + \dfrac{x^{\frac{1}{2}} + y^{\frac{1}{2}}}{x y^{\frac{1}{2}} - y x^{\frac{1}{2}}}\right)$
$= \dfrac{x^{\frac{3}{2}} y^{\frac{1}{2}} - xy - xy + x^{\frac{1}{2}} y^{\frac{3}{2}} + x^{\frac{3}{2}} y^{\frac{1}{2}} + xy + xy + x^{\frac{1}{2}} y^{\frac{3}{2}}}{\left(x y^{\frac{1}{2}} + y x^{\frac{1}{2}}\right) \left(x y^{\frac{1}{2}} - y x^{\frac{1}{2}}\right)}$
$= \dfrac{2 x^{\frac{3}{2}} y^{\frac{1}{2}} + 2 x^{\frac{1}{2}} y^{\frac{3}{2}}}{x^2 y - y^2 x}$
$= \dfrac{2 x^{\frac{1}{2}} y^{\frac{1}{2}} \left(x + y\right)}{xy \left(x - y\right)}$
$= \dfrac{2 x^{\frac{-1}{2}} y^{\frac{-1}{2}} \left(x + y\right)}{x - y}$ $\;\;\; \cdots \; (2)$
In view of $(2)$, expression $(1)$ becomes
$\left[\dfrac{2 x^{\frac{-1}{2}} y^{\frac{-1}{2}} \left(x + y\right)}{\left(x - y\right)}\right] \times \dfrac{x^{\frac{3}{2}} y^{\frac{1}{2}}}{\left(x + y\right)} - \dfrac{2y}{x - y}$
$= \dfrac{2x}{x - y} - \dfrac{2y}{x - y}$
$= \dfrac{2 \left(x - y\right)}{x - y}$
$= 2$