Explanation Given: The function is f ( x ) = cos x with the interval ( 0 , π / 2 ) . Concept used: The half range cosine series of a function is defined in the interval ( 0 , L ) is as follows: f ( x ) = a 0 2 + ∑ n = 1 ∞ a n cos n π x L ..... ( 1 ) where a n = 2 L ∫ 0 L f ( x ) cos n π x L d x , n ∈ ℕ a 0 = 2 L ∫ 0 L f ( x ) d x The half range sine series of a function is defined in the interval ( 0 , L ) is as follows: f ( x ) = ∑ n = 1 ∞ b n sin n π x L ...... ( 2 ) where b n = 2 L ∫ 0 L f ( x ) sin n π x L d x , n ∈ ℕ Calculation: Substitute f ( x ) = cos x and L = π 2 in the formula for a 0 as follows: a 0 = 2 L ∫ 0 L f ( x ) d x = 2 π 2 ∫ 0 π 2 cos x d x ( L = π 2 , f ( x ) = cos x ) Further, solve the above equation as follows: a 0 = 4 π [ sin x ] 0 π 2 = 4 π [ sin π 2 − sin 0 ] = 4 π Therefore, the value for a 0 is 4 π . Similarly, calculate the value for a n as follows: a n = 2 L ∫ 0 L f ( x ) cos n π x L d x = 2 π 2 ∫ 0 π 2 ( cos x ) cos n π x π 2 d x = 4 π ∫ 0 π 2 ( cos x ) ( cos 2 n x ) d x = 2 π ∫ 0 π 2 2 ( cos x ) ( cos 2 n x ) d x Further, solve the above expression as follows: a n = 2 π ∫ 0 π 2 ( cos ( 2 n + 1 ) x + cos ( 2 n − 1 ) x ) d x = 2 π [ sin ( 2 n + 1 ) x 2 n + 1 + sin ( 2 n − 1 ) x 2 n − 1 ] 0 π 2 = 2 π [ sin ( 2 n + 1 ) ( π 2 ) 2 n + 1 + sin ( 2 n − 1 ) ( π 2 ) 2 n − 1 ] = 2 π [ 1 2 n + 1 sin ( n π + π 2 ) + 1 2 n − 1 sin ( n π − π 2 ) ] Further, solve the above expression as follows: a n = 2 π [ 1 2 n + 1 cos n π − 1 2 n − 1 cos n π ] = 2 π [ ( 2 n − 1 ) − ( 2 n + 1 ) ( 2 n + 1 ) ( 2 n − 1 ) ] cos n π = 2 π [ − 2 4 n 2 − 1 ] ( − 1 ) n = 4 ( − 1 ) n π ( 1 − 4 n 2 ) Therefore, the value for a n is 4 ( − 1 ) n π ( 1 − 4 n 2 )

Differential Equations with Boundary-Value Problems

9th Edition

ISBN: 9781337632515

Author: Dennis G. Zill

Publisher: Cengage Learning US

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Chapter 11.3, Problem 27E

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The half-range cosine and sine expansion of the given function.

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Chapter 11 Solutions

Differential Equations with Boundary-Value Problems

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...

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The half-range cosine and sine expansion of the given function.

Solution Summary: The author describes the half-range cosine and sine expansion of a given function.

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The half-range cosine and sine expansion of the given function.

Chapter 11 Solutions