Program Explanation Given information: The system of differential equations is shown below. { x ′ 1 = 9 x 1 − 8 x 2 x ′ 2 = 6 x 1 − 5 x 2 Explanation: The system of differential equations can be written in x ′ = A x form as, x ′ = [ 9 − 8 6 − 5 ] x Where the coefficient matrix A is described as, A = [ 9 − 8 6 − 5 ] So, obtain the characteristic equation of matrix A to obtain the eigenvalues as, | 9 − λ − 8 6 − 5 − λ | = 0 ( 9 − λ ) ( − 5 − λ ) + 48 = 0 λ 2 − 4 λ + 3 = 0 Further, solve the above equation and obtain the eigenvalues as, λ 1 = 1 or λ 2 = 3 Now, obtain the eigenvector corresponding to λ 1 = 1 as follows: [ 9 − 1 − 8 6 − 5 − 1 ] [ a b ] = [ 0 0 ] [ 8 − 8 6 − 6 ] [ a b ] = [ 0 0 ] [ 8 a − 8 b 6 a − 6 b ] = [ 0 0 ] From the comparison of corresponding elements from both sides and simplification, it is obtained that a = b . For simplest vector, consider 1 for a and therefore b = 1 , and the eigenvector corresponding to λ 1 = 1 is written as, v 1 = [ 1 1 ] Similarly, obtain the eigenvector corresponding to λ 2 = 3 as follows: [ 9 − 3 − 8 6 − 5 − 3 ] [ a b ] = [ 0 0 ] [ 6 − 8 6 − 8 ] [ a b ] = [ 0 0 ] [ 6 a − 8 b 6 a − 8 b ] = [ 0 0 ] From the comparison of corresponding elements from both sides and simplification, it is obtained that 3 a = 4 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 13P

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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 12P

Chapter 5.6, Problem 14P

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1. Consider the linear equation system Ax = b with A = ( -9.8 19.8 -9.9 19.9 3) and b= = ( 1080 ) . In the following, you may use that the inverse of the matrix A is given by A-¹ = 1 = ( 19.9 -19.8 9.9 -9.8 8). (a) What is the solution x of Ax = b? (b) Let xx denote the solution of the perturbed equation (A+SA)(x+dx) = b for ||A|| 0.01. Show that the relative error is bounded by ||dx||∞ ||x||∞ ≤1.

2.2. Consider the following matrix Y and matrix Z. Each column represents a particular meat industry. industry for beef, pork and chicken: [0.2 0.3 0.21 Y= 0.4 0.1 0.3 [0.3 0.5 0.2] [150] Z= 200 [210] For industries for beef, pork and chicken determine the total demand given by matrix Y and matrix Z

1.2. Find a matrix A for which a1,1 = a2,2 = a3,3 = 1, but elimination (with no row exchanges) produces two negative pivots (the first pivot is 1).

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