Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 8.8, Problem 26E

The Bessel functions of order $v = n + 1 / 2$ , $n$ any integer, are related to the spherical Bessel functions. Use relation (33) and the results of Problem 25 to show that such Bessel functions can be represented in terms of $sin x$ , $cos x$ , and powers of $x$ . Demonstrate this by determining a closed form for $J − 3 / 2 ( x )$ and $J 5 / 2 ( x )$ .

$J v + 1 ( x ) = 2 v x J v ( x ) − J v − 1 ( x )$ , $( 33 )$

25. Show that

$J 1 / 2 ( x ) = ( 2 / π x ) 1 / 2 sin x$ and

$J − 1 / 2 ( x ) = ( 2 / π x ) 1 / 2 cos x$ .

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Chapter 8.8, Problem 25E

Chapter 8.8, Problem 27E

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Fundamentals of Differential Equations and Boundary Value Problems

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Solution Summary: The author explains that the Bessel functions of order v=n+1/2, n any integer, can be represented in terms of mathrmsinx

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