Program Explanation Given information: The system of differential equations is shown below. { x ′ 1 = 5 x 1 − 3 x 2 x ′ 2 = 2 x 1 Explanation: The system of differential equations can be written in x ′ = A x form as, x ′ = [ 5 − 3 2 0 ] x Where the coefficient matrix A is described as, A = [ 5 − 3 2 0 ] So, obtain the characteristic equation of matrix A to obtain the eigenvalues as, | 5 − λ − 3 2 0 − λ | = 0 ( 5 − λ ) ( 0 − λ ) + 6 = 0 λ 2 − 5 λ + 6 = 0 Further, solve the above equation and obtain the eigenvalues as, λ 1 = 3 or λ 2 = 2 Now, obtain the eigenvector corresponding to λ 1 = 3 as follows: [ 5 − 3 − 3 2 0 − 3 ] [ a b ] = [ 0 0 ] [ 2 − 3 2 − 3 ] [ a b ] = [ 0 0 ] [ 2 a − 3 b 2 a − 3 b ] = [ 0 0 ] From the comparison of corresponding elements from both sides and simplification, it is obtained that 2 a = 3 b . For simplest vector, consider 3 for a and therefore b = 2 , and the eigenvector corresponding to λ 1 = 3 is written as, v 1 = [ 3 2 ] Similarly, obtain the eigenvector corresponding to λ 2 = 2 as follows: [ 5 − 2 − 3 2 0 − 2 ] [ a b ] = [ 0 0 ] [ 3 − 3 2 − 2 ] [ a b ] = [ 0 0 ] [ 3 a − 3 b 2 a − 2 b ] = [ 0 0 ] From the comparison of corresponding elements from both sides and simplification, it is obtained that a = b

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 5.6, Problem 11P

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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 10P

Chapter 5.6, Problem 12P

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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 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