Explanation Given: The Function is defined as follows: f ( x ) = x 4 , − 1 < x < 1 Definition used: The Fourier Legendre series of a function f ( x ) on the interval − 1 < x < 1 is defined as follows: f ( x ) = ∑ n = 0 ∞ C n P n ( x ) , where the coefficient C i is defined as follows: C n = 2 n + 1 2 ∫ − 1 1 f ( x ) P n ( x ) d x Calculation: For the given function, calculate C 0 as follows: C 0 = 2 ( 0 ) + 1 2 ∫ − 1 1 x 4 ( 1 ) d x ( P 0 ( x ) = 1 , f ( x ) = x 4 , n = 0 ) = 1 2 [ x 5 5 ] − 1 1 = 1 2 [ 1 5 5 − ( − 1 ) 5 5 ] = 1 2 [ 2 5 ] C 0 = 1 5 Similarly, calculate C 1 as follows: C 1 = 2 ( 1 ) + 1 2 ∫ − 1 1 x 4 ( x ) d x ( P 1 ( x ) = x , f ( x ) = x 4 , n = 1 ) = 3 2 ∫ − 1 1 x 5 d x = 3 2 ( 0 ) ( ∫ − 1 1 x 5 d x = 0 , x 5 is odd function ) = 0 Similarly, calculate C 2 as follows: C 2 = 2 ( 2 ) + 1 2 ∫ − 1 1 x 4 ( 3 x 2 − 1 2 ) d x ( P 2 ( x ) = 3 x 2 − 1 2 , f ( x ) = x 4 , n = 2 ) = 5 2 ⋅ 1 2 ⋅ ∫ − 1 1 x 4 ( 3 x 2 − 1 ) d x = 5 4 ∫ − 1 1 ( 3 x 6 − x 4 ) d x = 5 4 [ 3 7 x 7 − 1 5 x 5 ] − 1 1 Further, solve the above expression as follows: C 2 = 5 4 [ 6 7 − 2 5 ] = 5 4 [ 16 35 ] = 4 7 Similarly, calculate C 3 as follows: C 3 = 2 ( 3 ) + 1 2 ∫ − 1 1 x 4 ( 5 x 3 − 3 x 2 ) d x

Differential Equations with Boundary-Value Problems

9th Edition

ISBN: 9781337632515

Author: Dennis G. Zill

Publisher: Cengage Learning US

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Chapter 11, Problem 22RE

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The Fourier-Legendre series for the function f(x)=x4,−1<x<1.

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Chapter 11, Problem 21RE

Chapter 11, Problem 23RE

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Chapter 11 Solutions

Differential Equations with Boundary-Value Problems

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The Fourier-Legendre series for the function f ( x ) = x 4 , − 1 < x < 1 .

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The Fourier-Legendre series for the function f(x)=x4,−1<x<1.

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Chapter 11 Solutions

The Fourier-Legendre series for the function f ( x ) = x 4 , − 1 &lt; x &lt; 1 .

Solution Summary: The author explains the Fourier-Legendre series for the function f(x)=x

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Chapter 11 Solutions

The Fourier-Legendre series for the function f ( x ) = x 4 , − 1 < x < 1 .