Let $y ( x )$ be the solution of the differential equation
$$x ^ { 2 } \frac { d y } { d x } + x y = x ^ { 2 } + y ^ { 2 } , \quad x > \frac { 1 } { e }$$
satisfying $y ( 1 ) = 0$. Then the value of $2 \frac { ( y ( e ) ) ^ { 2 } } { y \left( e ^ { 2 } \right) }$ is $\_\_\_\_$.