Use Power Series to Solve the Differential Equation $\left( x ^ { 2 } + 1 \right) y ^ { \prime \prime } + x y ^ { \prime } - y = 0$

Use power series to solve the differential equation.. $\left( x ^ { 2 } + 1 \right) y ^ { \prime \prime } + x y ^ { \prime } - y = 0$

A) $y ( x ) = c _ { 0 } \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( 2 n - 3 ) ! } { 2 ^ { 2 n - 2 } n ! ( n - 2 ) ! } x ^ { 2 n }$
B) $y ( x ) = c _ { 0 + } c _ { 1 } x + c _ { 0 } \frac { x ^ { 2 } } { 2 } + c _ { 0 } \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( 2 n - 3 ) ! } { 2 ^ { 2 n - 2 } n ! ( n - 2 ) ! } x ^ { 2 n + 1 }$
C) $y ( x ) = c _ { 0 + } c _ { 1 } x + c _ { 0 } \frac { x ^ { 2 } } { 2 } + c _ { 0 } \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( 2 n - 3 ) ! } { 2 ^ { 2 n - 2 } ( n - 2 ) ! } x ^ { 2 n }$
D) $y ( x ) = c _ { 0 + } c _ { 1 } x + c _ { 0 } \frac { x ^ { 2 } } { 2 } + c _ { 0 } \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( 2 n - 3 ) ! } { 2 ^ { 2 n - 2 } n ! ( n - 2 ) ! } x ^ { 2 n }$
E) $y ( x ) = c _ { 0 + } c _ { 1 } x + c _ { 0 } \frac { x ^ { 2 } } { 2 } + c _ { 0 } \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } ( 2 n ) ! } { 2 ^ { 2 n + 2 } n ! ( n - 2 ) ! } x ^ { 2 n + 1 }$

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