A Power Series Solution About $x = 0$ Of the Differential Equation

question 16

Multiple Choice

A power series solution about $x = 0$ of the differential equation $y ^ { \prime \prime } - y = 0$ is

• A) \(y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k } / ( 2 k ) ! + c _ { 1 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k + 1 } / ( 2 k + 1 ) !\)
• B) \(y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k } / ( 2 k ) + c _ { 1 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k + 1 } / ( 2 k + 1 )\)
• C) \(y = c _ { 0 } \sum _ { k - 0 } ^ { \infty } x ^ { 2 k } / ( 2 k ) ^ { 2 } + c _ { 1 } \sum _ { k - 0 } ^ { \infty } x ^ { 2 k + 1 } / ( 2 k + 1 ) ^ { 2 }\)
• D) \(y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k } / ( 2 k ) ! + c _ { 1 } \sum _ { k - 0 } ^ { \infty } x ^ { 2 k - 1 } / ( 2 k - 1 ) !\)
• E) \(y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k } / ( 2 k ) + c _ { 1 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k - 1 } / ( 2 k - 1 )\)

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