Explanation Given: The set of functions { 1 , x , x 2 , x 3 ⋯ } defined on the interval [ − 1 , 1 ] . Concept used: The linearly independent set of real valued functions that are continuous on an interval [ a , b ] is { f 0 ( x ) , f 1 ( x ) , f 2 ( x ) , … } . The inner product ( f n , ϕ n ) = ∫ a b f n ( x ) ϕ n ( x ) d x define the function in the set B ′ = { ϕ 0 ( x ) , ϕ 1 ( x ) , ϕ 2 ( x ) ⋯ } . According to Gram-Schmidt orthogonalization process, the set B ′ = { ϕ 0 ( x ) , ϕ 1 ( x ) , ϕ 2 ( x ) ⋯ } is a orthogonal set constructed form the orthogonal set { f 0 ( x ) , f 1 ( x ) , f 2 ( x ) , … } , if ϕ 0 ( x ) , ϕ 1 ( x ) … ϕ n ( x ) satisfy the condition as follows: ϕ 0 ( x ) = f 0 ( x ) ϕ 1 ( x ) = f 1 ( x ) − ( f 1 , ϕ 0 ) ‖ ϕ 0 ‖ 2 ϕ 0 ( x ) ϕ 2 ( x ) = f 2 ( x ) − ( f 2 , ϕ 0 ) ‖ ϕ 0 ‖ 2 ϕ 0 ( x ) − ( f 2 , ϕ 1 ) ‖ ϕ 1 ‖ 2 ϕ 1 ( x ) ϕ 3 ( x ) = f 3 ( x ) − ( f 3 , ϕ 0 ) ( ϕ 0 , ϕ 0 ) ϕ 0 ( x ) − ( f 3 , ϕ 1 ) ( ϕ 1 , ϕ 1 ) ϕ 1 ( x ) − ( f 3 , ϕ 2 ) ( ϕ 2 , ϕ 2 ) ϕ 2 ( x ) Calculation: Compare the set { 1 , x , x 2 , x 3 ⋯ } with { f 0 ( x ) , f 1 ( x ) , f 2 ( x ) , … } and assign the functions as follows: f 0 ( x ) = 1 f 1 ( x ) = x f 2 ( x ) = x 2 f 3 ( x ) = x 3 Now, form the orthogonal set by the use of Gram-Schmidt orthogonalization process. ϕ 0 ( x ) = f 0 ( x ) = 1 Therefore, the first function is ϕ 0 ( x ) = 1 . Similarly, obtain the function ϕ 1 ( x ) as follows: ϕ 1 ( x ) = f 1 ( x ) − ( f 1 , ϕ 0 ) ( ϕ 0 , ϕ 0 ) ϕ 0 ( x ) = x − ( x , 1 ) ( 1 , 1 ) ( 1 ) = x − ∫ − 1 1 x d x ∫ − 1 1 1 d x ( ∵ Interval [ − 1 , 1 ] ) = x − x 2 2 | − 1 1 x | − 1 1 Further, solve the above equation as follow: ϕ 1 ( x ) = x − 0 2 = x Therefore, the second function is ϕ 1 ( x ) = x

Differential Equations with Boundary-Value Problems

9th Edition

ISBN: 9781337632515

Author: Dennis G. Zill

Publisher: Cengage Learning US

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Chapter 11.1, Problem 27E

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The orthogonal function ϕ0(x), ϕ1(x), ϕ2(x) and ϕ3(x) in the orthogonal set B′.

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Chapter 11.1, Problem 26E

Chapter 11.1, Problem 28E

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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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The orthogonal function ϕ 0 ( x ) , ϕ 1 ( x ) , ϕ 2 ( x ) and ϕ 3 ( x ) in the orthogonal set B ′ .

Solution Summary: The author describes the linearly independent set of real valued functions that are continuous on an interval left[a,bright].

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The orthogonal function ϕ0(x), ϕ1(x), ϕ2(x) and ϕ3(x) in the orthogonal set B′.

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Chapter 11 Solutions