Program Explanation Given information: The system of differential equations is shown below. { x ′ 1 = 6 x 1 − 6 x 2 x ′ 2 = 4 x 1 − 4 x 2 Explanation: The system of differential equations can be written in x ′ = A x form as, x ′ = [ 6 − 6 4 − 4 ] x Where the coefficient matrix A is described as, A = [ 6 − 6 4 − 4 ] So, obtain the characteristic equation of matrix A to obtain the eigenvalues as, | 6 − λ − 6 4 − 4 − λ | = 0 ( 6 − λ ) ( − 4 − λ ) + 24 = 0 λ 2 − 2 λ = 0 λ ( λ − 2 ) = 0 Further, solve the above equation and obtain the eigenvalues as, λ 1 = 0 or λ 2 = 2 Now, obtain the eigenvector corresponding to λ 1 = 0 as follows: [ 6 − 0 − 6 4 − 4 − 0 ] [ a b ] = [ 0 0 ] [ 6 − 6 4 − 4 ] [ a b ] = [ 0 0 ] [ 6 a − 6 b 4 a − 4 b ] = [ 0 0 ] From the comparison of corresponding elements from both sides and simplification, it is obtained that b = a . For simplest vector, consider 1 for a and therefore b = 1 , and the eigenvector corresponding to λ 1 = 0 is written as, v 1 = [ 1 1 ] Similarly, obtain the eigenvector corresponding to λ 2 = 2 as follows: [ 6 − 2 − 6 4 − 4 − 2 ] [ a b ] = [ 0 0 ] [ 4 − 6 4 − 6 ] [ a b ] = [ 0 0 ] [ 4 a − 6 b 4 a − 4 b ] = [ 0 0 ] From the comparison of corresponding elements from both sides and simplification, it is obtained that 2 a = 3 b

5th Edition

ISBN: 9780135988510

Author: Edwards

Publisher: PEARSON CO

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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.

Question

Chapter 5.6, Problem 10P

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Program Description: Purpose of the problem is to obtain value of an exponential matrix eAt for the system x′=Ax .

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Chapter 5.6, Problem 9P

Chapter 5.6, Problem 11P

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Chapter 5 Solutions

EP DIFF.EQUAT.+BOUND.VALUE,...UPD.-ACC.

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 value of an exponential matrix e A t for the system x ′ = A x .

Solution Summary: The author explains that the problem is to obtain value of an exponential matrix eAt for the system of differential equations.

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Program Description: Purpose of the problem is to obtain value of an exponential matrix eAt for the system x′=Ax .

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Chapter 5 Solutions