Explanation Procedure used: Linear independence theorem. A set of functions is said to be independent if their Wronskian is nonzero. If the functions are f 1 and f 2 . Thus, W ( f 1 , f 2 ) = | f 1 f 2 f ′ 1 f ′ 2 | W ( f 1 , f 2 ) ≠ 0 Calculation: Consider the two functions f 1 ( x ) = x 2 and f 2 ( x ) = x | x | . Since, the function f 2 ( x ) = x | x | has two definitions. On the interval ( − ∞ , 0 ) , f 2 = − x 2 and for the interval ( ∞ , 0 ) , f 2 = x 2 . Thus, the Wronskian on the interval ( − ∞ , 0 )

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 4, Problem 11RE

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To find: An interval over which the two set of two functions f1(x)=x2 and f2(x)=x|x| is linearly independent. Then give an interval over which the set consisting of f1 and f2 is linearly dependent.

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Chapter 4, Problem 10RE

Chapter 4, Problem 12RE

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Chapter 4 Solutions

Differential Equations with Boundary-Value Problems (MindTap Course List)

Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - Given that y=c1+c2x2 is a two-parameter family of...Ch. 4.1 - Find two members of the family of solutions in...Ch. 4.1 - Prob. 7E Ch. 4.1 - Use the general solution of x + 2x = 0 given in...Ch. 4.1 - In Problems 9 and 10 find an interval centered...Ch. 4.1 - In Problems 9 and 10 find an interval centered...

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An interval over which the two set of two functions f 1 ( x ) = x 2 and f 2 ( x ) = x | x | is linearly independent. Then give an interval over which the set consisting of f 1 and f 2 is linearly dependent.

Solution Summary: The author explains the linear independence theorem, where a set of functions is said to be independent if their Wronskian is nonzero.

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To find: An interval over which the two set of two functions f1(x)=x2 and f2(x)=x|x| is linearly independent. Then give an interval over which the set consisting of f1 and f2 is linearly dependent.

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Chapter 4 Solutions