Program Explanation Given information: The coefficient matrix A for homogeneous system x ′ = A x is written below. A = [ 0 1 1 0 ] Explanation: First show that A 2 n = I by the principle of mathematical induction method as follows: Consider n = 1 and the statement become A 2 = I . A 2 = A A = [ 0 1 1 0 ] [ 0 1 1 0 ] = [ 1 0 0 1 ] Therefore, the equation A 2 n = I is true for n = 1 . Now, assume that A 2 k = I for any positive integer k and prove for the statement A 2 ( k + 1 ) = I as, A 2 ( k + 1 ) = A 2 k + 2 = A 2 k A 2 = ( I ) ( I ) = I Therefore, the statement A 2 k = I is true for all positive integer n . Now, post multiply the equation A 2 k = I by matrix A as, A 2 k A = I A A 2 k + 1 = A Therefore, the statement A 2 k + 1 = A is true for all positive integer n . Now, evaluate the matrix e A t as follows: e A t = I + A t + A 2 t 2 2 ! + A 3 t 3 3 ! + A 4 t 4 4 ! + ⋯ + A 2 n t 2 n ( 2 n ) ! + A 2 n + 1 t 2 n + 1 ( 2 n + 1 ) ! + ⋯ = I + A t + I t 2 2 ! + A t 3 3 ! + I t 4 4 ! + ⋯ + I t 2 n ( 2 n ) ! + A t 2 n + 1 ( 2 n + 1 ) ! + ⋯ = I ( 1 + t 2 2 ! + ⋯ + t 2 n ( 2 n ) ! + ⋯ ) + A ( t + t 3 3 ! + ⋯ + t 2 n + 1 ( 2 n + 1 ) ! + ⋯ ) = I cosh t + A sinh t Therefore, the solution of the homogeneous equation is written as, x = e A t c = ( I cosh t + A sinh t ) [ c 1 c 2 ] = ( [ cosh t 0 0 cosh t ] + [ 0 sinh t sinh t 0 ] ) [ c 1 c 2 ] = [ cosh t sinh t sinh t cosh t ] [ c 1 c 2 ] Further, solve the equation as shown below

5th Edition

ISBN: 9780135904206

Author: Edwards

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 5.6, Problem 33P

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Program Description: Purpose of the problem is to verify that general solution of x′=Ax found by the eigenvalue method is equivalent to the general solution found by using the fact that eAt=Icosht+Asinht for matrix A=[0110] .

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Chapter 5.6, Problem 32P

Chapter 5.6, Problem 34P

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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 verify that general solution of x ′ = A x found by the eigenvalue method is equivalent to the general solution found by using the fact that e A t = I cosh t + A sinh t for matrix A = [ 0 1 1 0 ] .

Solution Summary: The author explains how the equation A2n=I is true for n=1.

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Program Description: Purpose of the problem is to verify that general solution of x′=Ax found by the eigenvalue method is equivalent to the general solution found by using the fact that eAt=Icosht+Asinht for matrix A=[0110] .

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Chapter 5 Solutions