[tex]\dfrac{x}{y} = 2 \Rightarrow \dfrac{y}{x} = \dfrac{1}{2}[/tex]
[tex]\dfrac{x^2+y^2}{xy} = \dfrac{x^2}{xy}+\dfrac{y^2}{xy} = \dfrac{x}{y}+\dfrac{y}{x} = {^{^{^{\displaystyle 2)}}}} 2+\dfrac{1}{2} = \dfrac{4+1}{2} = \boxed{\dfrac{5}{2}}[/tex]
