Program Explanation Given information: The differential equation is x ′ = A x and the initial condition is x ( 0 ) = [ 4 2 ] and the matrix A = [ − 5 17 − 8 7 ] . Explanation: The characteristic equation for the matrix A is | A − λ I | = 0 . The characteristic equation | A − λ I | = 0 can be expressed as, | A − λ I | = 0 | − 5 − λ 17 − 8 7 − λ | = 0 ( 5 − λ ) ( 7 − λ ) + 136 = 0 λ 2 − 2 λ + 101 = 0 It can be seen that the resultant characteristic equation is quadratic equation. Therefore, the eigenvalue of the matrix can be found by the quadratic rule. λ = − b ± b 2 − 4 a c 2 a λ = 2 ± 20 i 2 λ = 1 ± 10 i Therefore, the complex conjugate eigenvalue of λ are λ = 1 + 10 i , λ = 1 − 10 i . The equation for the eigenvector v associated with eigenvalue λ 1 = 1 + 10 i is as follows, [ A − λI ] v = 0 [ A − ( 1 + 10 i ) ] v = 0 [ − 5 − ( 1 + 10 i ) 17 − 8 7 − ( 1 + 10 i ) ] [ a b ] = 0 [ − 6 − 10 i 17 − 8 − 6 − 10 i ] [ a b ] = 0 Now solve the equation we get, a = 6 − 10 i , b = 8 Therefore, the associated eigenvector with λ = 1 + 10 i is v = [ 6 − 10 i 8 ] T The solution x ( t ) for λ = 1 + 10 i can be calculated as, x ( t ) = v e ( 1 + 10 i ) t = [ 6 − 10 i 8 ] e ( 1 + 10 i ) t = [ 6 − 10 i 8 ] e t ( cos 10 t + i sin 10 t ) = [ 6 − 10 i ( cos 10 t + i sin 10 t ) 8 ( cos 10 t + i sin 10 t ) ] e t Now simplify the further solution

5th Edition

ISBN: 8220100802454

Author: 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 5.3, Problem 40P

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Program Description: Purpose of the problem is to show that the general solution of the equation x′=Ax is x1(t)=et(4cos10t+sin10t), x2(t)=et(2cos10t−2sin10t) .

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Chapter 5.3, Problem 39P

Chapter 5.4, Problem 1P

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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 show that the general solution of the equation x ′ = A x is x 1 ( t ) = e t ( 4 cos 10 t + sin 10 t ) , x 2 ( t ) = e t ( 2 cos 10 t − 2 sin 10 t ) .

Solution Summary: The author explains that the general solution of the equation xprime=Ax is a quadratic equation.

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Program Description: Purpose of the problem is to show that the general solution of the equation x′=Ax is x1(t)=et(4cos10t+sin10t), x2(t)=et(2cos10t−2sin10t) .

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Chapter 5 Solutions