Program Explanation Given information: The differential equation is x ′ 1 = x 1 + 2 x 2 + 2 x 3 , x ′ 2 = 2 x 1 + 7 x 2 + x 3 and x ′ 3 = 2 x 1 + x 2 + 7 x 3 . Explanation: Consider matrix form of the system of equation as A. A x = [ 1 2 2 2 7 1 2 1 7 ] x The characteristics equation of the coefficient matrix is, [ 1 − λ 2 2 2 7 − λ 1 2 1 7 − λ ] = 0 ( 1 − λ ) [ 7 − λ 1 1 7 − λ ] − 2 [ 2 1 2 7 − λ ] + 2 [ 2 7 − λ 2 1 ] = 0 − λ 3 + 15 λ 2 − 54 λ=0 − λ ( λ 2 − 15 λ + 54 ) = 0 Therefore, the Eigen values of the matrix can be obtained as, − λ ( λ − 6 ) ( λ − 9 ) = 0 − λ ( λ − 6 ) ( λ − 9 ) = 0 λ = 0 or λ = 6 or λ = 9 The Eigen vector of the coefficient matrix A is in the form as, [ 1 − λ 2 2 2 7 − λ 1 2 1 7 − λ ] [ a b c ] = [ 0 0 0 ] The associated Eigen vector with the matrix A is v = [ a b c ] T . Substitute 9 for λ in coefficient matrix, [ 1 − 9 2 2 2 7 − 9 1 2 1 7 − 9 ] [ a 1 b 1 c 1 ] = [ 0 0 0 ] [ − 8 2 2 2 − 2 1 2 1 − 2 ] [ a 1 b 1 c 1 ] = [ 0 0 0 ] The scalar equation are shown below, − 8 a 1 + 2 b 1 + 2 c 1 = 0 2 a 1 − 2 b 1 + c 1 = 0 2 a 1 + b 1 − 2 c 1 = 0 After solving the equation the value of a 1 , b 1 and c 1 can be obtained as, a 1 = 1 b 1 = 2 c 1 = 2 Therefore, the associated Eigen vector is shown below. v 1 = [ 1 2 2 ] Substitute 6 for λ in coefficient matrix, [ 1 − 6 2 2 2 7 − 6 1 2 1 7 − 6 ] [ a 2 b 2 c 2 ] = [ 0 0 0 ] [ − 5 2 2 2 1 1 2 1 1 ] [ a 2 b 2 c 2 ] = [ 0 0 0 ] The scalar equation are shown below, − 5 a 2 + 2 b 2 + 2 c 2 = 0 2 a 2 + b 2 + c 2 = 0 2 a 2 + b 2 + c 2 = 0 After solving the equation the value of a 2 , b 2 and c 2 can be obtained as, a 2 = 0 b 2 = 1 c 2 = − 1 Therefore, the associated Eigen vector is shown below

5th Edition

ISBN: 9780321974235

Author: Calvis

Publisher: PEARSON CUSTOM PUB.(CONSIGNMENT)

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Problems on Growing Functions

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Chapter 5.2, Problem 18P

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Program Description:Purpose of problem is to obtain the general solution of x′1=x1+2x2+2x3,x′2=2x1+7x2+x3 and x′3=2x1+x2+7x3 .

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Chapter 5.2, Problem 17P

Chapter 5.2, Problem 19P

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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 problem is to obtain the general solution of x ′ 1 = x 1 + 2 x 2 + 2 x 3 , x ′ 2 = 2 x 1 + 7 x 2 + x 3 and x ′ 3 = 2 x 1 + x 2 + 7 x 3 .

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

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Program Description:Purpose of problem is to obtain the general solution of x′1=x1+2x2+2x3,x′2=2x1+7x2+x3 and x′3=2x1+x2+7x3 .

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Chapter 5 Solutions