i = ( Find the eigenvalues and eigenvectors of the coefficient matrix A. 31 X₁ 7₁ I₁ dt and ₂: -17 -52 8 23, 18 -B . in Note: When Webwork checks whether eigenvectors and eigenvalues are correct, ALL eigenvalues and eigenvectors are evaluated at once. So individual eigenvectors and eigenvalues won't be marked correct unless all the eigenpairs are correct. Sketch a phase portrait. Solve the system given initial conditions x₁(0)=0, x₂(0) = -1.
i = ( Find the eigenvalues and eigenvectors of the coefficient matrix A. 31 X₁ 7₁ I₁ dt and ₂: -17 -52 8 23, 18 -B . in Note: When Webwork checks whether eigenvectors and eigenvalues are correct, ALL eigenvalues and eigenvectors are evaluated at once. So individual eigenvectors and eigenvalues won't be marked correct unless all the eigenpairs are correct. Sketch a phase portrait. Solve the system given initial conditions x₁(0)=0, x₂(0) = -1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![### Differential Equations: Eigenvalues and Eigenvectors
Consider the following system:
\[
\frac{d}{dt} \vec{x} = \begin{pmatrix} -17 & 8 \\ -52 & 23 \end{pmatrix} \vec{x}
\]
**Find the eigenvalues and eigenvectors of the coefficient matrix \( A \).**
#### Eigenvalues and Eigenvectors
\[
\lambda_1 = \_\_\_\_, \quad \vec{v}_1 = \begin{bmatrix} \_\_\_\_ \\ \_\_\_\_ \end{bmatrix}, \quad \text{and} \quad \lambda_2 = \_\_\_\_, \quad \vec{v}_2 = \begin{bmatrix} \_\_\_\_ \\ \_\_\_\_ \end{bmatrix}
\]
**Note:** When Webwork checks whether eigenvectors and eigenvalues are correct, all eigenvalues and eigenvectors are evaluated at once. Therefore, individual eigenvectors and eigenvalues won't be marked as correct unless all the eigenpairs are correct.
#### Phase Portrait
Sketch a phase portrait.
#### System with Initial Conditions
Solve the system given initial conditions \( x_1(0) = 0 \) and \( x_2(0) = -1 \).
\[
x_1 = \_\_\_\_
\]
\[
x_2 = \_\_\_\_
\]
**Graphs and Diagrams:**
No graphs or diagrams are provided in this exercise. If a phase portrait is to be sketched, it should display the trajectory of the system in the phase space, illustrating how the system evolves over time from chosen initial conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F216015d9-daff-4e98-b3d1-7f3f5202e820%2Fb3a8b112-66a0-4f3a-8b71-fba818b2113c%2Fogdxgl_processed.png&w=3840&q=75)
Transcribed Image Text:### Differential Equations: Eigenvalues and Eigenvectors
Consider the following system:
\[
\frac{d}{dt} \vec{x} = \begin{pmatrix} -17 & 8 \\ -52 & 23 \end{pmatrix} \vec{x}
\]
**Find the eigenvalues and eigenvectors of the coefficient matrix \( A \).**
#### Eigenvalues and Eigenvectors
\[
\lambda_1 = \_\_\_\_, \quad \vec{v}_1 = \begin{bmatrix} \_\_\_\_ \\ \_\_\_\_ \end{bmatrix}, \quad \text{and} \quad \lambda_2 = \_\_\_\_, \quad \vec{v}_2 = \begin{bmatrix} \_\_\_\_ \\ \_\_\_\_ \end{bmatrix}
\]
**Note:** When Webwork checks whether eigenvectors and eigenvalues are correct, all eigenvalues and eigenvectors are evaluated at once. Therefore, individual eigenvectors and eigenvalues won't be marked as correct unless all the eigenpairs are correct.
#### Phase Portrait
Sketch a phase portrait.
#### System with Initial Conditions
Solve the system given initial conditions \( x_1(0) = 0 \) and \( x_2(0) = -1 \).
\[
x_1 = \_\_\_\_
\]
\[
x_2 = \_\_\_\_
\]
**Graphs and Diagrams:**
No graphs or diagrams are provided in this exercise. If a phase portrait is to be sketched, it should display the trajectory of the system in the phase space, illustrating how the system evolves over time from chosen initial conditions.
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