Program Explanation Given information: The differential equation is x ′ 1 = 5 x 1 + 5 x 2 + 2 x 3 , x ′ 2 = − 6 x 1 − 6 x 2 − 5 x 3 and x ′ 3 = 6 x 1 + 6 x 2 + 5 x 3 . Explanation: Consider matrix form of the system of equation as A. A x = [ 5 5 2 − 6 − 6 − 5 6 6 5 ] x The characteristics equation of the coefficient matrix is, [ 5 − λ 5 2 − 6 − 6 − λ − 5 6 6 5 − λ ] = 0 ( 5 − λ ) [ − 6 − λ − 5 6 5 − λ ] − 5 [ − 6 − 5 6 5 − λ ] + 1 [ − 6 − 6 − λ 6 6 ] = 0 λ 3 − λ 2 +5λ 2 +5λ + 18 λ = 0 λ 3 − 4λ 2 + 13 λ = 0 Therefore, the Eigen values of the matrix can be obtained as, λ ( λ 2 − 4 λ+13 ) = 0 ( λ ) ( λ + 2 + 3 i ) ( λ+ 2 − 3 i ) = 0 λ = 0 or λ = 2 + 3 i or λ = 2 − 3 i The Eigen vector of the coefficient matrix A is in the form as, [ 5 − λ 5 2 − 6 − 6 − λ − 5 6 6 5 − λ ] [ a b c ] = [ 0 0 0 ] The associated Eigen vector with the matrix Ais v = [ a b c ] T . Substitute 0 for λ in coefficient matrix, [ 5 − 0 5 2 − 6 − 6 − 0 − 5 6 6 5 − 0 ] [ a 1 b 1 c 1 ] = [ 0 0 0 ] [ 5 5 2 − 6 − 6 − 5 6 6 5 ] [ a 1 b 1 c 1 ] = [ 0 0 0 ] The scalar equation are shown below, 5 a 1 + 5 b 1 + 2 c 1 = 0 − 6 a 1 − 6 b 1 − 5 c 1 = 0 6 a 1 + 6 b 1 + 5 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 = 1 c 1 = 0 Therefore, the associated Eigen vector is shown below. v 1 = [ − 1 1 0 ] Substitute 2 + 3 i for λ in coefficient matrix, [ 5 − 2 − 3 i 5 2 − 6 − 6 − 2 − 3 i − 5 6 6 5 − 2 − 3 i ] [ a 2 b 2 c 2 ] = [ 0 0 0 ] [ 3 − 3 i 5 2 − 6 − 8 − 3 i − 5 6 6 3 − 3 i ] [ a 2 b 2 c 2 ] = [ 0 0 0 ] The scalar equation are shown below, ( 3 − 3 i ) a 2 + 5 b 2 + 2 c 2 = 0 − 6 a 2 − ( 8 + 3 i ) b 2 − 5 c 2 = 0 6 a 2 + 6 b 2 + ( 3 − 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 + i b 2 = − 2 c 2 = 2 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

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 5.2, Problem 25P

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Program Description:Purpose of problem is to obtain the general solution of x′1=5x1+5x2+2x3,x′2=−6x1−6x2−5x3 and x′3=6x1+6x2+5x3 by Eigen values method.

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Chapter 5.2, Problem 24P

Chapter 5.2, Problem 26P

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Is the function f(x) continuous at x = 1? (x) 7 6 5 4 3 2 1 0 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -71 Select the correct answer below: The function f(x) is continuous at x = 1. The right limit does not equal the left limit. Therefore, the function is not continuous. The function f(x) is discontinuous at x = 1. We cannot tell if the function is continuous or discontinuous.

Question Is the function f(x) shown in the graph below continuous at x = -5? f(z) 7 6 5 4 2 1 0 -10 -6 -5 -4 1 0 2 3 5 7 10 -1 -2 -3 -4 -5 Select the correct answer below: The function f(x) is continuous. The right limit exists. Therefore, the function is continuous. The left limit exists. Therefore, the function is continuous. The function f(x) is discontinuous. We cannot tell if the function is continuous or discontinuous.

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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 = 5 x 1 + 5 x 2 + 2 x 3 , x ′ 2 = − 6 x 1 − 6 x 2 − 5 x 3 and x ′ 3 = 6 x 1 + 6 x 2 + 5 x 3 by Eigen values method.

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

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Program Description:Purpose of problem is to obtain the general solution of x′1=5x1+5x2+2x3,x′2=−6x1−6x2−5x3 and x′3=6x1+6x2+5x3 by Eigen values method.

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Chapter 5 Solutions