In the Previous Problem, a Series Solution Corresponding to the Indicial

In the previous problem, a series solution corresponding to the indicial root $r = 1 / 2$ is $y = x ^ { 1 / 2 } \left\{ 1 + \sum _ { k = 1 } ^ { \infty } c _ { k } x ^ { k } \right\}$ , where

A) $c _ { k } = ( - 2 ) ^ { k } / [ k \mid 3 \cdot 5 \cdot 7 \cdots ( 2 k - 3 ) ]$
B) $c _ { k } = ( - 2 ) ^ { k } / [ k \mid 1 \cdot 3 \cdot 5 \cdots ( 2 k - 3 ) ]$
C) $c _ { k } = - 2 ^ { k } / [ k \mid 5 \cdot 7 \cdot 9 \cdots ( 2 k + 1 ) ]$
D) $c _ { k } = ( - 2 ) ^ { k } / [ k ! ( 2 k + 3 ) ! ]$
E) $c _ { k } = ( - 2 ) ^ { k } / [ k \mid 5 \cdot 7 \cdot 9 \cdots ( 2 k + 3 ) ]$

Correct Answer:

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