Answer
$\displaystyle \frac{-2}{x+y}$ , $\; x\neq y$
Work Step by Step
When one denominator is the opposite, or additive inverse of the other,
first multiply either rational expression by $\displaystyle \frac{-1}{-1}$
to obtain a common denominator.
$\displaystyle \frac{2y}{x^{2}-y^{2}}+\frac{2x}{y^{2}-x^{2}} \cdot \displaystyle \frac{-1}{-1}= \displaystyle \frac{2y}{x^{2}-y^{2}}+\frac{-2x}{-(y^{2}-x^{2})}$
$=$ $\displaystyle \frac{2y}{x^{2}-y^{2}}+\frac{-2x}{x^{2}-y^{2}} $
... To add/subtract rational expressions with the same denominator,
add/subtract numerators and place the sum/difference over the common denominator.
$=\displaystyle \frac{2y-2x}{x^{2}-y^{2}}$
... factor both ... recognize a difference of squares
$=\displaystyle \frac{-2(x-y)}{(x+y)(x-y)}$
... common factors cancel,
= $\displaystyle \frac{-2}{x+y}$ , $\; x\neq y$
