Explanation Given: The Legendre polynomial P n ( x ) is given by, P n ( x ) = 2 − n ∑ m = 0 [ n / 2 ] ( − 1 ) m ( 2 n − 2 m ) ! ( n − m ) ! m ! ( n − 2 m ) ! x n − 2 m ... ( 1 ) Calculation: To obtain the first five Legendre polynomials, substitute 0 , 1 , 2 , 3 , and 4 for n in equation ( 1 ) . For n = 0 , P 0 ( x ) = 2 0 ∑ m = 0 0 ( − 1 ) m ( 2 ( 0 ) − 2 m ) ! ( 0 − m ) ! m ! ( 0 − 2 m ) ! x 0 − 2 m = 1 For n = 1 , P 1 ( x ) = 2 − 1 ∑ m = 0 0 ( − 1 ) m ( 2 ( 1 ) − 2 m ) ! ( 1 − m ) ! m ! ( 1 − 2 m ) ! x 1 − 2 m = 1 2 ( − 1 ) 0 ( 2 ) ! ( 1 ) ! 0 ! ( 1 ) ! x 1 = x For n = 2 , P 2 ( x ) = 2 − 2 ∑ m = 0 1 ( − 1 ) m ( 2 ( 2 ) − 2 m ) ! ( 2 − m ) ! m ! ( 2 − 2 m ) ! x 2 − 2 m = 1 4 ( ( − 1 ) 0 ( 4 − 2 ( 0 ) ) ! ( 2 − 0 ) ! 0 ! ( 2 − 2 ( 0 ) ) ! x 2 − 2 ( 0 ) + ( − 1 ) 1 ( 4 − 2 ( 1 ) ) ! ( 2 − 1 ) ! 1 ! ( 2 − 2 ( 1 ) ) ! x 2 − 2 ( 1 ) ) = 1 4 ( ( 4 ) ! ( 2 ) ! 0 ! ( 2 ) ! x 2 + ( − 1 ) ( 2 ) ! ( 1 ) ! 1 ! ( 0 ) ! x 0 ) = 1 4 ( 6 x 2 − 2 ) P 2 ( x ) = 3 2 x 2 − 1 2 For n = 3 ,

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 8.8, Problem 29E

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The first five Legendre polynomials.

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The first five Legendre polynomials.

Solution Summary: The author calculates the first five Legendre polynomials by substituting 0,1,2,3, and 4 for n in the equation (1).

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