(a) Explanation It is given that t x ′ = ( 5 4 3 4 3 4 5 4 ) x . Let A = ( 5 4 3 4 3 4 5 4 ) . Substitute A = ( 5 4 3 4 3 4 5 4 ) in det ( A − r I ) = 0 and obtain the eigen values as shown below: | [ 5 4 3 4 3 4 5 4 ] − r [ 1 0 0 1 ] | = 0 | 5 4 − r 3 4 3 4 5 4 − r | = 0 ⋯ ⋯ ( 1 ) ( 5 4 − r ) ( 5 4 − r ) − 9 16 = 0 r = 1 2 , 2 Thus, r = 1 2 , 2 . Substitute r = 1 2 in equation (1) and obtain the eigen vector as follows: [ 5 4 − r 3 4 3 4 5 4 − r ] = [ 5 4 − ( 1 2 ) 3 4 3 4 5 4 − ( 1 2 ) ] = [ 3 4 3 4 3 4 3 4 ] ∼ [ 1 1 1 1 ] Multiply [ − 1 1 1 − 1 ] [ x 1 x 2 ] and obtain the equations as follows: x 1 − x 2 = 0 Thus, the corresponding eigen vector associated with r = 1 2 is obtained as v 1 = [ 1 − 1 ] . Substitute r = 2 in equation (1) and obtain the eigen vector as follows: [ 5 4 − r 3 4 3 4 5 4 − r ] = [ 5 4 − ( 2 ) 3 4 3 4 5 4 − ( 2 ) ] = [ − 3 4 3 4 3 4 − 3 4 ] ∼ [ 1 1 1 1 ] Multiply [ − 3 4 3 4 3 4 − 3 4 ] [ x 1 x 2 ] and obtain the equations as follows: 3 4 x 1 + 3 4 x 2 = 0 Thus, the corresponding eigen vector associated with r = 2 is obtained as v 2 = [ 1 1 ] (b) To determine To plot: The directional field and few trajectories of the given system of equation.

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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Chapter 7.5, Problem 6P

(a)

To determine

The general solution of the system of equation and describe the behavior of the solution as t→∞.

(b)

To determine

To plot: The directional field and few trajectories of the given system of equation.

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Chapter 7.5, Problem 5P

Chapter 7.5, Problem 7P

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The general solution of the system of equation and describe the behavior of the solution as t → ∞ .