A Power Series Solution About $x = 0$ Of the Differential Equation

Multiple Choice

A power series solution about $x = 0$ of the differential equation $y ^ { \prime \prime } + y = 0$ is

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