Explanation Consider the given linear system in problem 22. d x d t = − 6 x + 5 y , d y d t = − 5 x + 4 y Equation above can be written in the matrix form as follow. X ′ = ( − 6 5 − 5 4 ) X Now, the eigenvalues of the coefficient matrix is evaluated as follows. Here, I is the identity matrix of order 2 × 2 . | A − λ I | = 0 | − 6 − λ 5 − 5 4 − λ | = 0 ...... ( 1 ) The above determinant is expanded as shown below. ( − 6 − λ ) ( 4 − λ ) + 25 = 0 − 24 + 2 λ + λ 2 + 25 = 0 λ 2 + 2 λ + 1 = 0 ( λ + 1 ) 2 = 0 λ = − 1 , − 1 Thus, the eigenvalues are λ 1 = − 1 , λ 2 = − 1 . The eigenvector corresponding to the eigenvalue λ 1 = − 1 is K = ( 1 1 ) The eigenvector corresponding to the eigenvalue λ 2 = − 1 is P = ( 0 1 5 ) General solution of the given linear system is written as follows. X ( t ) = c 1 ( 1 1 ) e − t + c 2 [ ( 1 1 ) t e − t + ( 0 1 5 ) e − t ] From the above expression x 1 ( t ) , x 2 ( t ) are obtained. Thus, the solution to the given linear system is x 1 ( t ) = c 1 e − t + c 2 t e − t , x 2 ( t ) = c 1 e − t + c 2 ( t e − t + 1 5 e − t ) . Assume x 1 ( t ) = x and x 2 ( t ) = y . Let c 1 = 1 and c 2 = 0 in the above obtained solution. Then, x = e − t , y = e − t . Eliminate the parameters and thus the equation obtained is y = x , x > 0 . Also when c 1 = − 1 and c 2 = 0 , the equation obtained will be y = x , x < 0 . Now, enter the values in the online graphing calculator (Geogbra-phase portrait for homogenous linear first order system differential equation) as in the matrix form. Some calculators will take the values as − 6 x + 5 y and − 5 x + 4 y . Phase portrait of the given linear system is shown below. Phase portrait above represents all 2 × 2 homogenous linear system X ′ = A X . The darker lines represent the half-line trajectories y = x and the bold arrow represents the eigenvectors of the given system

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

11th Edition

ISBN: 9781305965720

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 8.2, Problem 34E

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To sketch: The phase portraits for the given systems in problems 22 and 23 and also to find any half-line trajectories and include these lines in the phase portrait.

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Chapter 8.2, Problem 33E

Chapter 8.2, Problem 35E

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Chapter 8 Solutions

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

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The phase portraits for the given systems in problems 22 and 23 and also to find any half-line trajectories and include these lines in the phase portrait.

Solution Summary: The author illustrates the phase portraits and half-line trajectories of linear systems in problems 22 and 23.

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To sketch: The phase portraits for the given systems in problems 22 and 23 and also to find any half-line trajectories and include these lines in the phase portrait.

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Chapter 8 Solutions