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

Multiple Choice

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

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