Program Explanation Given information: The differential equation is d x 1 d t = 3 x 1 + x 3 , d x 2 d t = 9 x 1 − x 2 + 2 x 3 and d x 2 d t = − 9 x 1 + 4 x 2 − x 3 . The initial condition is x 1 ( 0 ) = 0 , x 2 ( 0 ) = 0 and x 3 ( 0 ) = 17 . Explanation: Consider matrix form of the system of equation as A. A = [ 3 0 1 9 − 1 2 − 9 4 − 1 ] The characteristics equation of the coefficient matrix is, [ 3 − λ 0 1 9 − 1 − λ 2 − 9 4 − 1 − λ ] = 0 ( 3 − λ ) [ − 1 − λ 2 4 − 1 − λ ] − 0 [ 9 2 − 9 − 1 − λ ] + 1 [ 9 − 1 − λ − 9 4 ] = 0 λ 3 +λ 2 +4λ + 6 = 0 Therefore, the Eigen values of the matrix can be obtained as, λ 3 +λ 2 +4λ + 6 = 0 λ = 3 or λ = − 1 + i or λ = − 1 − i The Eigen vector of the coefficient matrix A is in the form as, [ 3 − λ 0 1 9 − 1 − λ 2 − 9 4 − 1 − λ ] [ a b c ] = [ 0 0 0 ] The associated Eigen vector with the matrix Ais v = [ a b c ] T . Substitute 3 for λ in coefficient matrix, [ 3 − 3 0 1 9 − 1 − 3 2 − 9 4 − 1 − 3 ] [ a 1 b 1 c 1 ] = [ 0 0 0 ] [ 0 0 1 9 − 4 2 − 9 4 − 4 ] [ a 1 b 1 c 1 ] = [ 0 0 0 ] The scalar equation are shown below, 9 a 1 − 4 b 1 − 2 c 1 = 0 − 9 a 1 + 4 b 1 − 4 c 1 = 0 After solving the equation the value of a 1 , b 1 and c 1 can be obtained as, a 1 = 4 b 1 = 9 c 1 = 0 Therefore, the associated Eigen vector is shown below. v 1 = [ 4 9 0 ] Substitute − 1 + i for λ in coefficient matrix, [ 3 + 1 − i 0 1 9 − 1 + 1 − i 2 − 9 4 − 1 + 1 − i ] [ a 2 b 2 c 2 ] = [ 0 0 0 ] [ 4 − i 0 1 9 − i 2 − 9 4 − i ] [ a 2 b 2 c 2 ] = [ 0 0 0 ] The scalar equation are shown below, ( 4 − i ) a 2 + c 2 = 0 9 a 2 − ( i ) b 2 + 2 c 2 = 0 − 9 a 2 + 4 b 2 − i c 2 = 0 After solving the equation the value of a 2 , b 2 and c 2 can be obtained as, a 2 = 1 b 2 = 2 − i c 2 = − 4 + i Therefore, the associated Eigen vector is shown below

5th Edition

ISBN: 9780321974235

Author: Calvis

Publisher: PEARSON CUSTOM PUB.(CONSIGNMENT)

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A naïve solution for the above-stated problem is to evaluate all of the possible combinations of different pieces of varying size to find the maximum revenue-generating combination. Although this method seems the easiest and quick to practice, it has exp…

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Chapter 5.2, Problem 26P

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Program Description:Purpose of problem is to obtain the particular solution of the system of first order differential equation dx1dt=3x1+x3,dx2dt=9x1−x2+2x3 and dx2dt=−9x1+4x2−x3 .

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Chapter 5.2, Problem 25P

Chapter 5.2, Problem 27P

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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 particular solution of the system of first order differential equation d x 1 d t = 3 x 1 + x 3 , d x 2 d t = 9 x 1 − x 2 + 2 x 3 and d x 2 d t = − 9 x 1 + 4 x 2 − x 3 .

Solution Summary: The author explains that the problem is to obtain the particular solution of the system of first order differential equation.

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Program Description:Purpose of problem is to obtain the particular solution of the system of first order differential equation dx1dt=3x1+x3,dx2dt=9x1−x2+2x3 and dx2dt=−9x1+4x2−x3 .

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Chapter 5 Solutions