Explanation Given: The graph of the Bessel functions J 0 ( x ) and J 1 ( x ) is given by, Approach: Rolle’s theorem There exists at least one c in open interval ( α , α + 1 ) such that J ′ 1 ( c ) = 0 . The recurrence relation involving Bessel function is, d d x [ x v J v ( x ) ] = x v J v − 1 ( x ) ... ( 1 ) Calculation: Let the two consecutive roots be α and α + 1 . As J 1 ( x ) is continuous in [ α , α + 1 ] , J 1 ( x ) is derivable in ( α , α + 1 ) J 1 ( α ) = J 1 ( α + 1 ) = 0 Substitute 1 for v in ( 1 ) . d d x [ x J 1 ( x ) ] = x J 0 ( x ) x J ′ 1 ( x ) + J 1 ( x ) = x J 0 ( x ) … ( 2 ) Substitute α for x in ( 2 )

Fundamentals of Differential Equations [With CDROM] - 7th Edition

7th Edition

ISBN: 9780321410481

Author: Saff, Edward B., Snider, Arthur David, Nagle, R. Kent

Publisher: Addison Wesley

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Chapter 8.8, Problem 28E

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There is a root J0(x) between tow consecutive positive roots of J1(x).

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Chapter 8.8, Problem 27E

Chapter 8.8, Problem 29E

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Fundamentals of Differential Equations [With CDROM] - 7th Edition

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There is a root J 0 ( x ) between tow consecutive positive roots of J 1 ( x ) .

Solution Summary: The author explains that the graph of the Bessel functions J_0(x) is given by, Approach: Rolle’s theorem.

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To show:

There is a root J0(x) between tow consecutive positive roots of J1(x).

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Chapter 8 Solutions

There is a root J 0 ( x ) between tow consecutive positive roots of J 1 ( x ) .

Solution Summary: The author explains that the graph of the Bessel functions J_0(x) is given by, Approach: Rolle’s theorem.

Concept explainers

Videos

To show: There is a root J0(x) between tow consecutive positive roots of J1(x).

Want to see the full answer?

Chapter 8 Solutions

To show:

There is a root J0(x) between tow consecutive positive roots of J1(x).