(a) Explanation The given set is { sin π x 2 L , sin 3 π x 2 L , sin 5 π x 2 L , ⋯ } and the interval is [ 0 , L ] . The pattern of the angles is observed as π x , 3 π x , 5 π x ⋯ , ( 2 n + 1 ) π x . Use the rule 2 sin C ⋅ sin D = cos ( C − D ) − cos ( C + D ) to obtain the product. When m ≠ n , obtain the product of elements of the set as follows: I = ∫ 0 L sin ( 2 n + 1 ) π x 2 L sin ( 2 m + 1 ) π x 2 L d x = 2 L π ∫ 0 π 2 sin { ( 2 n + 1 ) t } sin { ( 2 m + 1 ) t } d t ( t = π x 2 L implies d t = π 2 L d x ) = L π ∫ 0 π 2 2 sin { ( 2 n + 1 ) t } sin { ( 2 m + 1 ) t } d t = L π ∫ 0 π 2 [ cos { ( 2 n + 1 − 2 m − 1 ) t } − cos { ( 2 n + 1 + 2 m + 1 ) t } (b) To determine The norm of each function of the set { sin π x 2 L , sin 3 π x 2 L , sin 5 π x 2 L , ⋯ } .

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Question

Chapter 11, Problem 12RE

(a)

To determine

To show: The given set is orthogonal on the interval [0,L].

(b)

To determine

The norm of each function of the set {sinπx2L,sin3πx2L,sin5πx2L,⋯}.

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Chapter 11, Problem 11RE

Chapter 11, Problem 13RE

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Chapter 11 Solutions

Differential Equations with Boundary-Value Problems (MindTap Course List)

Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - Prob. 6E Ch. 11.1 - Prob. 7E Ch. 11.1 - In Problems 712 show that the given set of...Ch. 11.1 - In Problems 712 show that the given set of...Ch. 11.1 - In Problems 712 show that the given set of...

Knowledge Booster

The given set is orthogonal on the interval [ 0 , L ] .

Solution Summary: The author explains that the given set is orthogonal on the interval left[0,Lright] and the pattern of the angles is observed.

To show: The given set is orthogonal on the interval [0,L].

The norm of each function of the set {sinπx2L,sin3πx2L,sin5πx2L,⋯}.

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Chapter 11 Solutions