Explanation Given: The functions are, f 1 ( x ) = 0 , f 2 ( x ) = x , f 3 ( x ) = e x . Definition used: Linearly dependence or independence: “A set of functions f 1 ( x ) , f 2 ( x ) , … , f n ( x ) is said to be linearly dependent on an interval I if there exists constant c 1 , c 2 , … , c n , not all zero, such that c 1 f 1 ( x ) + c 2 f 2 ( x ) + ...... + c n f n ( x ) = 0 for every x in the interval. If the set of functions is not linearly dependent on the interval, it is said to be linearly independent”. Wronskian functions: “Suppose each of the functions f 1 ( x ) , f 2 ( x ) , … , f n ( x ) possesses at least n − 1 derivatives. The determinant, W ( f 1 , f 2 , … , f n ) = | f 1 f 2 ⋯ f n f 1 f 2 ⋯ f n ⋮ ⋮ ⋮ f 1 ( n − 1 ) f 2 ( n − 1 ) ⋯ f n ( n − 1 ) | where the prime denotes the derivative, is called the Wronskian functions”. Theorem used: “Let f 1 ( x ) , f 2 ( x ) , … , f n ( x ) be n solutions of the homogenous linear n th-order differential equation on an interval I . Then the solutions is linearly independent on I if and only if W ( y 1 , y 2 , … , y n ) ≠ 0 for every x in the interval.” Calculation: Consider the set of functions are, f 1 ( x ) = 0 , f 2 ( x ) = x , f 3 ( x ) = e x . Obtain the Wronskian for the functions f 1 ( x ) = 0 , f 2 ( x ) = x , f 3 ( x ) = e x as follows. W ( f 1 , f 2 , f 3 ) = | f 1 f 2 f 3 f ′ 1 f ′ 2 f ′ 3 f ″ 1 f ″ 2 f ″ 3 | Compute the values of f ′ 1 and f ″ 1 as follows

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 3.1, Problem 16E

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Whether the set of functions f1(x)=0,f2(x)=x,f3(x)=ex is linearly independent or not in the interval (−∞,∞).

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Chapter 3.1, Problem 15E

Chapter 3.1, Problem 17E

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In Problems 11–20, for the given functions f and g. find: (a) (f° g)(4) (b) (g•f)(2) (c) (fof)(1) (d) (g ° g)(0) \ 11. f(x) = 2x; g(x) = 3x² + 1 12. f(x) = 3x + 2; g(x) = 2x² – 1 1 13. f(x) = 4x² – 3; g(x) = 3 14. f(x) = 2x²; g(x) = 1 – 3x² 15. f(x) = Vx; 8(x) = 2x 16. f(x) = Vx + 1; g(x) = 3x %3D 1. 17. f(x) = |x|; g(x) = 18. f(x) = |x – 2|: g(x) x² + 2 2 x + 1 x² + 1 19. f(x) = 3 8(x) = Vĩ 20. f(x) = x³/2; g(x) = X + 1'

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Chapter 3 Solutions

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Whether the set of functions f 1 ( x ) = 0 , f 2 ( x ) = x , f 3 ( x ) = e x is linearly independent or not in the interval ( − ∞ , ∞ ) .

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Whether the set of functions f1(x)=0,f2(x)=x,f3(x)=ex is linearly independent or not in the interval (−∞,∞).

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