Explanation Given: The function is f ( x ) = x + 1 , 0 < x < 1 . Definition used: The Fourier series expansion of a function f ( x ) for the interval ( 0 , L ) is as follows: f ( x ) = a 0 2 + ∑ n = 1 ∞ [ a n cos 2 n π x L + b n sin 2 n π x L ] , where a 0 = 2 L ∫ 0 L f ( x ) d x a n = 2 L ∫ 0 L f ( x ) cos ( 2 n π L x ) d x b n = 2 L ∫ 0 L f ( x ) sin ( 2 n π L x ) d x Calculation: Substitute f ( x ) = x + 1 , 0 < x < 1 and L = 1 in the formula for a 0 . a 0 = 2 L ∫ 0 L f ( x ) d x = 2 1 ∫ 0 1 ( x + 1 ) d x ( L = 1 , f ( x ) = x + 1 ) = 2 [ x 2 2 + x ] 0 1 = 2 [ 1 2 + 1 ] Further, solve the above expression as follows: a 0 = 2 ( 3 2 ) = 3 Similarly, calculate the value for a n as follows: a n = 2 L ∫ 0 L f ( x ) cos ( 2 n π L x ) d x = 2 1 ∫ 0 1 ( x + 1 ) cos ( 2 n π 1 x ) d x ( L = 1 , f ( x ) = x + 1 ) = 2 ∫ 0 1 ( x + 1 ) ( cos 2 n π x ) d x = 2 [ ( x + 1 ) ( 1 2 n π sin 2 n π x ) − ( 1 ) ( − 1 4 n 2 π 2 cos 2 n π x ) ] 0 1 Further, solve the above expression as follows: a n = 2 [ − ( 1 ) ( − 1 4 n 2 π 2 cos 2 n π ) + ( 1 ) ( − 1 4 n 2 π 2 cos 0 ) ] = 2 [ 1 4 n 2 π 2 ( cos 2 n π ) + ( 1 ) ( − 1 4 <

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 37E

To determine

The Fourier series expansion of the function f(x)=x+1,0<x<1.

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Chapter 11.3, Problem 36E

Chapter 11.3, Problem 38E

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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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