Explanation Given: The function is f ( x ) = { x , 0 < x < π 2 π − x , π 2 ≤ x < π . 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 < π 2 π − x , π 2 ≤ x < π and L = π in the formula for a 0 as follows: a 0 = 2 L ∫ 0 L f ( x ) d x = 2 π ∫ 0 π f ( x ) d x ( L = π ) = 2 π ( ∫ 0 π 2 x d x + ∫ π 2 π ( π − x ) d x ) = 2 π ( [ x 2 2 ] 0 π 2 + [ π x − x 2 2 ] π 2 π ) Further, solve the above equation as follows: a 0 = 2 π ( [ π 2 8 ] + [ π 2 − π 2 2 − π 2 2 + π 2 8 ] ) = 2 π ( π 2 8 + π 2 8 ) = 2 π ( π 2 4 ) = π 2 Therefore, the value for a 0 is π 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 π [ ∫ 0 π / 2 x ( cos n x ) d x + ∫ π / 2 π ( π − x ) ( cos n x ) d x ] ( L = π ) = 2 π ( [ ( x ) ( sin n x n ) − ( 1 ) ( − cos n x n 2 ) ] 0 π 2 + [ ( π − x ) ( sin n x n ) − ( − 1 ) ( − cos n x n 2 ) ] π 2 π ) = 2 π ( [ π 2 n sin n π 2 + 1 n 2 cos n π 2 − 1 n 2 ] + [ − 1 n 2 cos n π − π 2 n sin n π 2 + 1 n 2 cos n π 2 ] ) Further, solve the above expression as follows: a n = 2 π ( π 2 n sin n π 2 + 1 n 2 cos n π 2 − 1 n 2 − 1 n 2 cos n π − π 2 n sin n π 2 + 1 n 2 cos n π 2 ) = 2 π ( 2 n 2 cos n π 2 − 1 n 2 − 1 n 2 cos n π ) = 2 π ( 2 n 2 cos n π 2 − 1 n 2 − 1 n 2 ( − 1 ) n ) Further, solve the above expression as follows: a n = 2 π ( 2 n 2 cos n π 2 − 1 n 2 − 1 n 2 ( − 1 ) n ) = 2 π ( 2 cos n π 2 − 1 − ( − 1 ) n n 2 ) Now substitute the value of the coefficients a 0 and a n in ( 1 )

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 29E

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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

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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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