Program Explanation Given information: The system is x ′ = [ 5 − 1 1 1 3 0 − 3 2 1 ] x . The Eigen values is given as, λ = 3 , 3 , 3 Explanation: Consider matrix form of the system of equation as A . A = [ 5 − 1 1 1 3 0 − 3 2 1 ] The associated Eigen vector with the matrix A is v = [ a b c ] T . Substitute 3 for λ in coefficient matrix ( A − λ I ) v = 0 , [ 5 − 3 − 1 1 1 3 − 3 0 − 3 2 1 − 3 ] [ a 1 b 1 c 1 ] = [ 0 0 0 ] [ 2 − 1 1 1 0 0 − 3 2 − 2 ] [ a 1 b 1 c 1 ] = [ 0 0 0 ] The scalar equation are shown below, 2 a 1 − b 1 + c 1 = 0 a 1 = 0 − 3 a 1 + 2 b 1 − 2 c 1 = 0 Substitute 0 for a 1 in equation 2 a 1 − b 1 + c 1 = 0 as, 2 ( 0 ) − b 1 + c 1 = 0 b 1 = c 1 Consider 2 k as b 1 then the value of a 1 and c 1 can be obtained as, a 1 = 0 and c 1 = 2 k Now, the vector can be obtained as, [ a 1 b 1 c 1 ] = [ 0 2 k 2 k ] = [ 0 2 2 ] k Therefore, the associated Eigen vector is obtained as: v 1 = [ 0 2 2 ] The value of ( A − λ I ) 2 can be obtained as, ( A − λ I ) 2 = [ 2 − 1 1 1 0 0 − 3 2 − 2 ] [ 2 − 1 1 1 0 0 − 3 2 − 2 ] = [ 0 0 0 2 − 1 1 2 − 1 1 ] The value of ( A − λ I ) 3 can be obtained as, ( A − λ I ) 3 = [ 0 0 0 2 − 1 1 2 − 1 1 ] [ 2 − 1 1 1 0 0 − 3 2 − 2 ] = [ 0 0 0 0 0 0 0 0 0 ] Therefore, any vector v ≠ 0 is a solution of the equation ( A − 3 I ) 3 v = 0

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

5th Edition

ISBN: 9780321816252

Author: C. Henry Edwards, David E. Penney, David Calvis

Publisher: PEARSON

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Chapter 5.5, Problem 26P

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Program Description:Purpose of problem is to obtain the general solution of the system x′=[5−11130−321]x .

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Chapter 5.5, Problem 25P

Chapter 5.5, Problem 27P

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Chapter 5 Solutions

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

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 problem is to obtain the general solution of the system x ′ = [ 5 − 1 1 1 3 0 − 3 2 1 ] x .

Solution Summary: The author explains that the problem is to obtain the general solution of the system of equation as A. Substitute 3 for in coefficient matrix

Program Description:Purpose of problem is to obtain the general solution of the system x′=[5−11130−321]x .

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Chapter 5 Solutions