(a) Explanation Consider the given linear system. X ′ = ( 1 1 1 1 ) 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 | 1 − λ 1 1 1 − λ | = 0 ...... ( 1 ) The above determinant is expanded as shown below. ( 1 − λ ) 2 − 1 = 0 1 + λ 2 − 2 λ − 1 = 0 λ ( λ − 2 ) = 0 λ = 0 or λ = 2 Thus, the eigenvalues are λ 1 = 0 , λ 2 = 2 . Substitute λ 1 = 0 in (1) to evaluate the first eigenvector. ( 1 1 1 1 ) ( k 1 k 2 ) = ( 0 0 ) Reduce the above matrix on the LHS using row echelon form as shown below. ( 1 1 1 1 ) R 2 → R 2 − R 1 ( 1 1 0 0 ) The matrix obtained is written as shown below. ( 1 1 0 0 ) ( k 1 k 2 ) = ( 0 0 ) From the above matrix, the first row of the matrix means k 1 = − k 2 Assume k 2 = 1 , then the above equations will give k 1 = − 1 . Thus, the eigenvector corresponding to the eigenvalue λ 1 = 0 is K 1 = ( − 1 1 ) Substitute λ 2 = 2 in (1) to evaluate the second eigenvector. ( − 1 1 1 − 1 ) ( k 1 k 2 ) = ( 0 0 ) Reduce the above matrix on the LHS using row echelon form as shown below (b) To determine To solve: The given linear system, sketch the phase portrait of the given system and also to determine the significance of the line y = − x in the portrait.

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 53E

(a)

To determine

To solve: The given linear system, sketch the phase portrait of the given system and also to determine the significance of the line y=−x in the portrait.

(b)

To determine

To solve: The given linear system, sketch the phase portrait of the given system and also to determine the significance of the line y=−x in the portrait.

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Chapter 8.2, Problem 51E

Chapter 8.2, Problem 54E

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

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The given linear system, sketch the phase portrait of the given system and also to determine the significance of the line y = − x in the portrait.

Solution Summary: The author analyzes the linear system by sketching the phase portrait and determining the significance of the line y=-x in the portrait.

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To solve: The given linear system, sketch the phase portrait of the given system and also to determine the significance of the line y=−x in the portrait.

To solve: The given linear system, sketch the phase portrait of the given system and also to determine the significance of the line y=−x in the portrait.

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Chapter 8 Solutions