Consider the Differential Equation $x y ^ { \prime \prime } - x y ^ { \prime } + y = 0$

question 6

Multiple Choice

Consider the differential equation $x y ^ { \prime \prime } - x y ^ { \prime } + y = 0$ . The indicial equation is $r ( r - 1 ) = 0$ . The recurrence relation is $c _ { k + 1 } ( k + r + 1 ) + ( k + r ) - c _ { k } ( k + r - 1 ) = 0$ . A series solution corresponding to the indicial root $r = 0$ is

• A) \(y _ { 1 } = x\)
• B) \(y _ { 1 } = x ^ { 2 }\)
• C) \(y _ { 1 } = \sum _ { k = 0 } ^ { \infty } ( - 2 x ) ^ { k } / [ k ! ( - 1 ) \cdot 1 \cdot 3 \cdots ( 2 k - 1 ) ]\)
• D) \(y _ { 1 } = \sum _ { k = 0 } ^ { \infty } ( - 2 x ) ^ { k } / [ k ! ( 2 k - 3 ) ! ]\)
• E) \(y _ { 1 } = \sum _ { k = 0 } ^ { \infty } ( - 2 x ) ^ { k } / [ k ! 1 \cdot 3 \cdots ( 2 k - 3 ) ]\)

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