The Solution of the Initial-Value Problem $x ^ { 2 } y ^ { tt} + x y ^ { \prime } + x ^ { 2 } y = 0 , y ( 0 ) = 1 , y ^ { t } ( 0 ) = 0$

question 13

Multiple Choice

The solution of the initial-value problem $x ^ { 2 } y ^ { tt} + x y ^ { \prime } + x ^ { 2 } y = 0 , y ( 0 ) = 1 , y ^ { t } ( 0 ) = 0$ is called a Bessel function of order 0 . Solve the initial - value problem to find a power series expansion for the Bessel function.

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

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Definitions:

Cognitive Style

A characteristic mode of processing information, perceiving, and problem-solving, which can vary among individuals.

Cultural Background

The cultural environment and experiences that influence an individual's beliefs, values, and behaviors.

Expectation

A belief or anticipation about the outcome of an event or situation in the future.

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A tendency of the human perception system to see familiar objects as having standard shape, size, color, and location regardless of changes in the angle of perspective, distance, and lighting.