Using Power Series Methods, the Solution Of $x y ^ { \prime \prime } - x y ^ { \prime } + y = 0$

Using power series methods, the solution of $x y ^ { \prime \prime } - x y ^ { \prime } + y = 0$ is

A) $y = c _ { 0 } x + c _ { 1 } \left[ x \ln x - 1 + \sum _ { n = 2 } ^ { \infty } x ^ { n } / n ! \right]$
B) $y = c _ { 0 } x + c _ { 1 } \left[ x \ln x - 1 + \sum _ { n = 1 } ^ { \infty } x ^ { n } / ( n ! ( n + 1 ) ) \right]$
C) $y = c _ { 0 } x + c _ { 1 } \left[ x \ln x + \sum _ { n = 2 } ^ { \infty } x ^ { n } / ( n ! ( n - 1 ) ) \right]$
D) $y = c _ { 0 } x + c _ { 1 } \left[ x \ln x + \sum _ { n = 1 } ^ { \infty } x ^ { n } / ( n ! ( n - 1 ) ) \right]$
E) $y = c _ { 0 } x + c _ { 1 } \left[ x \ln x - 1 + \sum _ { n = 2 } ^ { \infty } x ^ { n } / ( n ! ( n - 1 ) ) \right]$

Correct Answer:

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