Explanation Given: The given function is f ( x ) = { x , 0 < x < 1 1 , 1 ≤ x < 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 ) = { x , 0 < x < 1 1 , 1 ≤ x < 2 and L = 2 in the formula for a 0 as follows: a 0 = 2 L ∫ 0 L f ( x ) d x = 2 2 ∫ 0 2 f ( x ) d x ( L = 2 ) = ( ∫ 0 1 ( x ) d x + ∫ 1 2 ( 1 ) d x ) = [ x 2 2 ] 0 1 + [ x ] 1 2 Further, solve the above equation as follows: a 0 = [ x 2 2 ] 0 1 + [ x ] 1 2 = 1 2 + 1 = 3 2 Therefore, the value for a 0 is 3 2 . 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 2 [ ∫ 0 1 ( x ) ( cos n π x 2 ) d x + ∫ 1 2 ( 1 ) ( cos n π x 2 ) d x ] ( L = 2 ) = [ ( x ) ( 2 n π sin n π x 2 ) − ( 1 ) ( − 4 n 2 π 2 cos n π x 2 ) ] 0 1 + [ 2 n π sin n π x 2 ] 1 2 = 2 n π sin n π 2 + 4 n 2 π 2 cos n π 2 − 4 n 2 π 2 − 2 n π sin n π 2 Further, solve the above expression as follows: a n = 4 n 2 π 2 cos n π 2 − 4 n 2 π 2 = 4 n 2 π 2 ( cos n π 2 − 1 ) Now substitute the value of the coefficients a 0 and a n in ( 1 ) . The half range cosine series of the given function is as follows: f ( x ) = a 0 2 + ∑ n = 1 ∞ a n cos n π x L = 3 4 + ∑ n = 1 ∞ a n cos n π x 2 ( a 0 = 3 2 , L = 2 ) = 3 4 + ∑ n = 1 ∞ 4 n 2 π 2 ( cos n π 2 − 1 ) cos n π x 2 ( a n = 4 n 2 π 2 ( cos n π 2 − 1 ) ) Hence, the half range cosine series of the given function is f ( x ) = 3 4 + ∑ n = 1 ∞ 4 n 2 π 2 ( cos n π 2 − 1 ) cos n π x 2

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

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 11.3, Problem 31E

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

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Chapter 11.3, Problem 30E

Chapter 11.3, Problem 32E

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Chapter 11 Solutions

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

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

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

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