Consider the Differential Equation $x y ^ { \prime \prime } - x y ^ { \prime } + y = 0$

Consider the differential equation $x y ^ { \prime \prime } - x y ^ { \prime } + y = 0$ . The indicial equation is $r ( r - 1 ) = 0$ . The recurrence relation is $c _ { k + 1 } ( k + r + 1 ) + ( k + r ) - c _ { k } ( k + r - 1 ) = 0$ . A series solution corresponding to the indicial root $r = 0$ is

A) $y _ { 1 } = x$
B) $y _ { 1 } = x ^ { 2 }$
C) $y _ { 1 } = \sum _ { k = 0 } ^ { \infty } ( - 2 x ) ^ { k } / [ k ! ( - 1 ) \cdot 1 \cdot 3 \cdots ( 2 k - 1 ) ]$
D) $y _ { 1 } = \sum _ { k = 0 } ^ { \infty } ( - 2 x ) ^ { k } / [ k ! ( 2 k - 3 ) ! ]$
E) $y _ { 1 } = \sum _ { k = 0 } ^ { \infty } ( - 2 x ) ^ { k } / [ k ! 1 \cdot 3 \cdots ( 2 k - 3 ) ]$