Consider the Differential Equation $2 x ^ { 2 } y ^ { \prime \prime } + 3 x y ^ { \prime } + ( 2 x - 1 ) y = 0$

Consider the differential equation $2 x ^ { 2 } y ^ { \prime \prime } + 3 x y ^ { \prime } + ( 2 x - 1 ) y = 0$ The indicial equation is $2 r ^ { 2 } + r - 1 = 0$ . The recurrence relation is $c _ { k } [ 2 ( k + r ) + ( k + r - 1 ) + 3 ( k + r ) - 1 ] + 2 c _ { k - 1 } = 0$ . A series solution corresponding to the indicial root $r = - 1$ is $y = x ^ { - 1 } \left[ 1 + \sum _ { k = 1 } ^ { \infty } c _ { k } x ^ { k } \right]$ , where

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