2013 AMC 12A Problem 8

Multiplying by $xy$ gives $x^2 y + 2y = xy^2 + 2x,$ so $$\begin{gathered} x^2 y - xy^2 - 2x + 2y \\ = (x - y)(xy - 2) \\ = 0. \end{gathered}$$ Since $x \ne y,$ it follows that $xy = 2.$ Thus, the correct answer is D.