Using Maclaurin Series, the General Series Solution, with the First $y ^ { \prime \prime } + 2 x y ^ { \prime } + 2 y = 0$

Multiple Choice

Using Maclaurin series, the general series solution, with the first three nonzero terms, of the differential equation $y ^ { \prime \prime } + 2 x y ^ { \prime } + 2 y = 0$ is

A) $y = A \left( 1 - x ^ { 2 } + \frac { x ^ { 4 } } { 2 } - \cdots \right) + B \left( x - \frac { 2 x ^ { 3 } } { 3 } + \frac { 4 x ^ { 5 } } { 15 } + \cdots \right)$
B) $y = A \left( 1 + x ^ { 2 } - \frac { x ^ { 4 } } { 2 } + \cdots \right) + B \left( x - \frac { 2 x ^ { 3 } } { 3 } + \frac { 4 x ^ { 5 } } { 15 } + \cdots \right)$
C) $y = A \left( 1 - x ^ { 2 } + \frac { x ^ { 4 } } { 2 } - \cdots \right) + B \left( x + \frac { 2 x ^ { 3 } } { 3 } - \frac { 4 x ^ { 5 } } { 15 } + \cdots \right)$
D) $y = A \left( 1 + x ^ { 2 } + \frac { x ^ { 4 } } { 2 } - \cdots \right) + B \left( x + \frac { 2 x ^ { 3 } } { 3 } + \frac { 4 x ^ { 5 } } { 15 } + \cdots \right)$
E) $y = A \left( 1 - x ^ { 2 } + \frac { x ^ { 4 } } { 2 } - \cdots \right) + B \left( x - \frac { 2 x ^ { 3 } } { 3 } + \frac { 4 x ^ { 5 } } { 15 } - \cdots \right)$