Program Explanation Given information: The value of each mass and spring constant is written below. m 1 = 1 , m 2 = 1 , m 3 = 1 k 1 = 2 , k 2 = 2 , k 3 = 2 , k 4 = 2 Explanation: The matrix differential equation for the given system is written as, M x ″ = K x Here the mass matrix M is written as, M = [ m 1 0 0 0 m 2 0 0 0 m 3 ] M = [ 1 0 0 0 1 0 0 0 1 ] Similarly the stiffness matrix K is written as, K = [ − ( k 1 + k 2 ) k 2 0 k 2 − ( k 2 + k 3 ) k 3 0 k 3 − ( k 3 + k 4 ) ] K = [ − 4 2 0 2 − 4 2 0 2 − 4 ] Therefore, matrix differential equation is written as, [ 1 0 0 0 1 0 0 0 1 ] x ″ = [ − 4 2 0 2 − 4 2 0 2 − 4 ] x Reduce the above differential equation in simplest form as, x ″ = [ 1 0 0 0 1 0 0 0 1 ] − 1 [ − 4 2 0 2 − 4 2 0 2 − 4 ] x x ″ = [ 1 0 0 0 1 0 0 0 1 ] [ − 4 2 0 2 − 4 2 0 2 − 4 ] x x ″ = [ − 4 2 0 2 − 4 2 0 2 − 4 ] x Compare the above equation with x ″ = A x , and obtain the matrix A as, A = [ − 4 2 0 2 − 4 2 0 2 − 4 ] Use calculator or matrix program to find the eigenvalues and corresponding eigenvectors for the matrix A and obtain the result as, λ 1 = − 4 ⇒ v 1 = [ − 1 0 1 ] T λ 2 = − 4 + 2 2 ⇒ v 2 = [ 1 2 1 ] T λ 3 = − 4 − 2 2 ⇒ v 3 = [ 1 − 2 1 ] T So, the displacement vector corresponding to ω 1 = 2 for λ 1 = − 4 is written as, x ( t ) = ( a 1 cos ω 1 t + b 1 sin ω 1 t ) v 1 x ( t ) = ( a 1 cos 2 t + b 1 sin 2 t ) [ − 1 0 1 ] x ( t ) =

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

5th Edition

ISBN: 9780321816252

Author: C. Henry Edwards, David E. Penney, David Calvis

Publisher: PEARSON

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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 5.4, Problem 13P

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Program Description: Purpose of the problem is to find the three natural frequencies of the system and obtain the ratio of amplitudes corresponding to each natural frequency.

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Chapter 5.4, Problem 12P

Chapter 5.4, Problem 14P

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Chapter 5 Solutions

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

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

Ch. 5.1 - Prob. 11P Ch. 5.1 - Prob. 12P Ch. 5.1 - Prob. 13P Ch. 5.1 - Prob. 14P Ch. 5.1 - Prob. 15P Ch. 5.1 - Prob. 16P Ch. 5.1 - Prob. 17P Ch. 5.1 - Prob. 18P Ch. 5.1 - Prob. 19P Ch. 5.1 - Prob. 20P Ch. 5.1 - Prob. 21P Ch. 5.1 - Prob. 22P Ch. 5.1 - Prob. 23P Ch. 5.1 - Prob. 24P Ch. 5.1 - Prob. 25P Ch. 5.1 - Prob. 26P Ch. 5.1 - Prob. 27P Ch. 5.1 - Prob. 28P Ch. 5.1 - Prob. 29P Ch. 5.1 - Prob. 30P Ch. 5.1 - Prob. 31P Ch. 5.1 - Prob. 32P Ch. 5.1 - Prob. 33P Ch. 5.1 - Prob. 34P Ch. 5.1 - Prob. 35P Ch. 5.1 - Prob. 36P Ch. 5.1 - Prob. 37P Ch. 5.1 - Prob. 38P Ch. 5.1 - Prob. 39P Ch. 5.1 - Prob. 40P Ch. 5.1 - Prob. 41P Ch. 5.1 - Prob. 42P Ch. 5.1 - Prob. 43P Ch. 5.1 - Prob. 44P Ch. 5.1 - Prob. 45P Ch. 5.2 - Prob. 1P Ch. 5.2 - Prob. 2P Ch. 5.2 - Prob. 3P Ch. 5.2 - Prob. 4P Ch. 5.2 - Prob. 5P Ch. 5.2 - Prob. 6P Ch. 5.2 - Prob. 7P Ch. 5.2 - Prob. 8P Ch. 5.2 - Prob. 9P Ch. 5.2 - Prob. 10P Ch. 5.2 - Prob. 11P Ch. 5.2 - Prob. 12P Ch. 5.2 - Prob. 13P Ch. 5.2 - Prob. 14P Ch. 5.2 - Prob. 15P Ch. 5.2 - Prob. 16P Ch. 5.2 - Prob. 17P Ch. 5.2 - Prob. 18P Ch. 5.2 - Prob. 19P Ch. 5.2 - Prob. 20P Ch. 5.2 - Prob. 21P Ch. 5.2 - Prob. 22P Ch. 5.2 - Prob. 23P Ch. 5.2 - Prob. 24P Ch. 5.2 - Prob. 25P Ch. 5.2 - Prob. 26P Ch. 5.2 - Prob. 27P Ch. 5.2 - Prob. 28P Ch. 5.2 - Prob. 29P Ch. 5.2 - Prob. 30P Ch. 5.2 - Prob. 31P Ch. 5.2 - Prob. 32P Ch. 5.2 - Prob. 33P Ch. 5.2 - Prob. 34P Ch. 5.2 - Prob. 35P Ch. 5.2 - Prob. 36P Ch. 5.2 - Prob. 37P Ch. 5.2 - Prob. 38P Ch. 5.2 - Prob. 39P Ch. 5.2 - Prob. 40P Ch. 5.2 - Prob. 41P Ch. 5.2 - Prob. 42P Ch. 5.2 - Prob. 43P Ch. 5.2 - Prob. 44P Ch. 5.2 - Prob. 45P Ch. 5.2 - Prob. 46P Ch. 5.2 - Prob. 47P Ch. 5.2 - Prob. 48P Ch. 5.2 - Prob. 49P Ch. 5.2 - Prob. 50P Ch. 5.3 - Prob. 1P Ch. 5.3 - Prob. 2P Ch. 5.3 - Prob. 3P Ch. 5.3 - Prob. 4P Ch. 5.3 - Prob. 5P Ch. 5.3 - Prob. 6P Ch. 5.3 - Prob. 7P Ch. 5.3 - Prob. 8P Ch. 5.3 - Prob. 9P Ch. 5.3 - Prob. 10P Ch. 5.3 - Prob. 11P Ch. 5.3 - Prob. 12P Ch. 5.3 - Prob. 13P Ch. 5.3 - Prob. 14P Ch. 5.3 - Prob. 15P Ch. 5.3 - Prob. 16P Ch. 5.3 - Prob. 17P Ch. 5.3 - Prob. 18P Ch. 5.3 - Prob. 19P Ch. 5.3 - Prob. 20P Ch. 5.3 - Prob. 21P Ch. 5.3 - Prob. 22P Ch. 5.3 - Prob. 23P Ch. 5.3 - Prob. 24P Ch. 5.3 - Prob. 25P Ch. 5.3 - Prob. 26P Ch. 5.3 - Prob. 27P Ch. 5.3 - Prob. 28P Ch. 5.3 - Prob. 29P Ch. 5.3 - Prob. 30P Ch. 5.3 - Prob. 31P Ch. 5.3 - Prob. 32P Ch. 5.3 - Prob. 33P Ch. 5.3 - Verify Eq. 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Program Description: Purpose of the problem is to find the three natural frequencies of the system and obtain the ratio of amplitudes corresponding to each natural frequency.

Solution Summary: The author explains that the problem is to find the three natural frequencies of the system and obtain the ratio of amplitudes corresponding to each natural frequency.

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Program Description: Purpose of the problem is to find the three natural frequencies of the system and obtain the ratio of amplitudes corresponding to each natural frequency.

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Chapter 5 Solutions