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

5th Edition

ISBN: 9780100802452

Author: Calvis

Publisher: YUZU

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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 9P

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

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Chapter 5.6, Problem 8P

Chapter 5.6, Problem 10P

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7. (S.15). Find the inverse A-1 of the matrix -2 -3 A = 0 2 -1 -1 -2 3

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.

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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(53) by substituting the expressions...Ch. 5.3 - Prob. 35P Ch. 5.3 - Prob. 36P Ch. 5.3 - Prob. 37P Ch. 5.3 - Prob. 38P Ch. 5.3 - Prob. 39P Ch. 5.3 - Prob. 40P Ch. 5.4 - Prob. 1P Ch. 5.4 - Prob. 2P Ch. 5.4 - Prob. 3P Ch. 5.4 - Prob. 4P Ch. 5.4 - Prob. 5P Ch. 5.4 - Prob. 6P Ch. 5.4 - Prob. 7P Ch. 5.4 - Prob. 8P Ch. 5.4 - Prob. 9P Ch. 5.4 - Prob. 10P Ch. 5.4 - Prob. 11P Ch. 5.4 - Prob. 12P Ch. 5.4 - Prob. 13P Ch. 5.4 - Prob. 14P Ch. 5.4 - Prob. 15P Ch. 5.4 - Prob. 16P Ch. 5.4 - Prob. 17P Ch. 5.4 - Prob. 18P Ch. 5.4 - Prob. 19P Ch. 5.4 - Prob. 20P Ch. 5.4 - Prob. 21P Ch. 5.4 - Prob. 22P Ch. 5.4 - Prob. 23P Ch. 5.4 - Prob. 24P Ch. 5.4 - Prob. 25P Ch. 5.4 - Prob. 26P Ch. 5.4 - Prob. 27P Ch. 5.4 - Prob. 28P Ch. 5.4 - Prob. 29P Ch. 5.5 - Prob. 1P Ch. 5.5 - Prob. 2P Ch. 5.5 - Prob. 3P Ch. 5.5 - Prob. 4P Ch. 5.5 - Prob. 5P Ch. 5.5 - Prob. 6P Ch. 5.5 - Prob. 7P Ch. 5.5 - Prob. 8P Ch. 5.5 - Prob. 9P Ch. 5.5 - Prob. 10P Ch. 5.5 - Prob. 11P Ch. 5.5 - Prob. 12P Ch. 5.5 - Prob. 13P Ch. 5.5 - Prob. 14P Ch. 5.5 - Prob. 15P Ch. 5.5 - Prob. 16P Ch. 5.5 - Prob. 17P Ch. 5.5 - Prob. 18P Ch. 5.5 - Prob. 19P Ch. 5.5 - Prob. 20P Ch. 5.5 - Prob. 21P Ch. 5.5 - Prob. 22P Ch. 5.5 - Prob. 23P Ch. 5.5 - Prob. 24P Ch. 5.5 - Prob. 25P Ch. 5.5 - Prob. 26P Ch. 5.5 - Prob. 27P Ch. 5.5 - Prob. 28P Ch. 5.5 - Prob. 29P Ch. 5.5 - Prob. 30P Ch. 5.5 - Prob. 31P Ch. 5.5 - Prob. 32P Ch. 5.5 - Prob. 33P Ch. 5.5 - Prob. 34P Ch. 5.5 - Prob. 35P Ch. 5.5 - Prob. 36P Ch. 5.6 - Prob. 1P Ch. 5.6 - Prob. 2P Ch. 5.6 - Prob. 3P Ch. 5.6 - Prob. 4P Ch. 5.6 - Prob. 5P Ch. 5.6 - Prob. 6P Ch. 5.6 - Prob. 7P Ch. 5.6 - Prob. 8P Ch. 5.6 - Prob. 9P Ch. 5.6 - Prob. 10P Ch. 5.6 - Prob. 11P Ch. 5.6 - Prob. 12P Ch. 5.6 - Prob. 13P Ch. 5.6 - Prob. 14P Ch. 5.6 - Prob. 15P Ch. 5.6 - Prob. 16P Ch. 5.6 - Prob. 17P Ch. 5.6 - Prob. 18P Ch. 5.6 - Prob. 19P Ch. 5.6 - Prob. 20P Ch. 5.6 - Prob. 21P Ch. 5.6 - Prob. 22P Ch. 5.6 - Prob. 23P Ch. 5.6 - Prob. 24P Ch. 5.6 - Prob. 25P Ch. 5.6 - Prob. 26P Ch. 5.6 - Prob. 27P Ch. 5.6 - Prob. 28P Ch. 5.6 - Prob. 29P Ch. 5.6 - Prob. 30P Ch. 5.6 - Prob. 31P Ch. 5.6 - Prob. 32P Ch. 5.6 - Prob. 33P Ch. 5.6 - Prob. 34P Ch. 5.6 - Prob. 35P Ch. 5.6 - Prob. 36P Ch. 5.6 - Prob. 37P Ch. 5.6 - Prob. 38P Ch. 5.6 - Prob. 39P Ch. 5.6 - Prob. 40P Ch. 5.7 - Prob. 1P Ch. 5.7 - Prob. 2P Ch. 5.7 - Prob. 3P Ch. 5.7 - Prob. 4P Ch. 5.7 - Prob. 5P Ch. 5.7 - Prob. 6P Ch. 5.7 - Prob. 7P Ch. 5.7 - Prob. 8P Ch. 5.7 - Prob. 9P Ch. 5.7 - Prob. 10P Ch. 5.7 - Prob. 11P Ch. 5.7 - Prob. 12P Ch. 5.7 - Prob. 13P Ch. 5.7 - Prob. 14P Ch. 5.7 - Prob. 15P Ch. 5.7 - Prob. 16P Ch. 5.7 - Prob. 17P Ch. 5.7 - Prob. 18P Ch. 5.7 - Prob. 19P Ch. 5.7 - Prob. 20P Ch. 5.7 - Prob. 21P Ch. 5.7 - Prob. 22P Ch. 5.7 - Prob. 23P Ch. 5.7 - Prob. 24P Ch. 5.7 - Prob. 25P Ch. 5.7 - Prob. 26P Ch. 5.7 - Prob. 27P Ch. 5.7 - Prob. 28P Ch. 5.7 - Prob. 29P Ch. 5.7 - Prob. 30P Ch. 5.7 - Prob. 31P Ch. 5.7 - Prob. 32P Ch. 5.7 - Prob. 33P Ch. 5.7 - Prob. 34P

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Program Description: Purpose of the problem is to obtain value ofanexponential matrix e A t for the system x ′ = A x .

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

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

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Chapter 5 Solutions