Program Explanation Given information: The matrix is A = [ 25 12 0 − 18 − 5 0 6 6 13 ] . Calculation: The given system of the differential equation x ′ = [ 25 12 0 − 18 − 5 0 6 6 13 ] x . The characteristic equation of the coefficient matrix is | A − λI | = 0 and it can be calculated as, | 25 − λ 12 0 − 18 − 5 − λ 0 6 6 13 − λ | = 0 ( 25 − λ ) ( − 5 − λ ) ( 13 − λ ) + 12 ⋅ 18 ( 13 − λ ) = 0 ( 13 − λ ) 2 ( 7 − λ ) λ 1 = 7 , λ 2 = 13 , λ 3 = 13 Therefore, the real eigenvalue of the coefficient matrix are λ 1 = 7 , λ 2 = 13 , λ 3 = 13 . Now substitute 7 for λ in the characteristic equation | A − λI | = 0 to obtain the value of a , b , c as, [ 25 − 7 12 0 − 18 − 5 − 7 0 6 6 13 − 7 ] [ a b c ] = [ 0 0 0 ] [ 18 12 0 − 18 − 12 0 6 6 6 ] [ a b c ] = [ 0 0 0 ] [ 18 a + 12 b − 18 a − 18 b 6 a + 6 b + 6 c ] = [ 0 0 0 ] Now compare the corresponding coefficient to obtain the value of a , b and c . 3 a = − 2 b , b = − 3 c Assume the value of c = s and s = 1 . Therefore, the associated eigenvector with λ 1 = 7 is v 1 = [ 2 − 3 1 ] T Now substitute 13 for λ in the characteristic equation | A − λI | = 0 to obtain the value of a , b , c as, [ 25 − 13 12 0 − 18 − 5 − 13 0 6 6 13 − 13 ] [ a b c ] = [ 0 0 0 ] [ 12 12 0 − 18 − 18 0 6 6 0 ] [ a b c ] = [ 0 0 0 ] [ 12 a + 12 b − 18 a − 18 b 6 a + 6 b ] = [ 0 0 0 ] Now compare the corresponding coefficient to obtain the value of a , b and c . 6 a + 6 b = 0 It an be observed that the value of c is nonzero

5th Edition

ISBN: 9780100802452

Author: Calvis

Publisher: YUZU

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Problems on Growing Functions

For a given set of functions f and g, the growth of functions states that the function g grows as fast as the function f. The growth rate of algorithms helps determine the algorithm's efficiency and compare its performance to other algorithms.

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Chapter 5.5, Problem 8P

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Program Description: Purpose of the problem is to obtain the general solution of the system x′=Ax .

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Chapter 5.5, Problem 7P

Chapter 5.5, Problem 9P

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Chapter 5 Solutions

EBK DIFFERENTIAL EQUATIONS

Ch. 5.1 - Let A=[2347] and B=[3451]. Find (a) 2A+3B; (b)...Ch. 5.1 - Prob. 2P Ch. 5.1 - Find AB and BA given A=[203415] and B=[137032].Ch. 5.1 - Prob. 4P Ch. 5.1 - Prob. 5P Ch. 5.1 - Prob. 6P Ch. 5.1 - Prob. 7P Ch. 5.1 - Prob. 8P Ch. 5.1 - Prob. 9P Ch. 5.1 - Prob. 10P

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Program Description: Purpose of the problem is to obtain the general solution of the system x ′ = A x .

Solution Summary: The author explains that the problem is to obtain the general solution of the system xprime =Ax.

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Program Description: Purpose of the problem is to obtain the general solution of the system x′=Ax .

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Chapter 5 Solutions