(a) Explanation Formula used: Spherical Bessel function of first kind: j n ( x ) = π 2 x J n + 1 / 2 ( x ) Recurrence relation: (i) 2 ν J ν ( x ) = x J ν + 1 ( x ) + x J ν − 1 ( x ) (ii) x J ′ ν ( x ) = − ν J ν ( x ) + x J ν − 1 ( x ) Bessel functions of order 1 2 and − 1 2 : J 1 / 2 ( x ) = 2 π x sin x and J − 1 / 2 ( x ) = 2 π x cos x . Calculation: Consider the spherical Bessel function of first kind. j n ( x ) = π 2 x J n + 1 / 2 ( x ) Substitute n = 1 and n = 2 in the above Bessel’s function. n = 1 , j 1 ( x ) = π 2 x J 3 / 2 ( x ) ...... ( 1 ) n = 2 , j 2 ( x ) = π 2 x J 5 / 2 ( x ) ...... ( 2 ) Now, use the recurrence relation (i) with ν = 1 / 2 . J 1 / 2 ( x ) = x J 3 / 2 ( x ) + x J − 1 / 2 ( x ) x J 3 / 2 ( x ) = J 1 / 2 ( x ) − x J − 1 / 2 ( x ) J 3 / 2 ( x ) = 1 x J 1 / 2 ( x ) − J − 1 / 2 ( x ) The value of J 3 / 2 ( x ) is calculated as follows. J 3 / 2 ( x ) = 1 x 2 π x sin x − 2 π x cos x = 2 π x ( sin x x − cos x ) Similarly, use the recurrence relation (i) with ν = 3 / 2 . 3 J 3 / 2 ( x ) = x J 5 / 2 ( x ) + x J 1 / 2 ( x ) x J 5 / 2 ( x ) = 3 J 3 / 2 ( x ) − x J 1 / 2 ( x ) J 5 / 2 ( x ) = 3 x J 3 / 2 ( x ) − J 1 / 2 ( x ) The value of J 3 / 2 ( x ) is obtained above (b) To determine To sketch: The graph of j 1 ( x ) and j 2 ( x ) in the same coordinate plane using graphing calculator.

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

11th Edition

ISBN: 9781305965720

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 6.4, Problem 33E

(a)

To determine

The spherical Bessel functions j1(x) and j2(x).

(b)

To determine

To sketch: The graph of j1(x) and j2(x) in the same coordinate plane using graphing calculator.

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Chapter 6.4, Problem 32E

Chapter 6.4, Problem 34E

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The spherical Bessel functions j 1 ( x ) and j 2 ( x ) .

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