Consider the following system: 22-(2) * 7 k dt Find the eigenvalues in terms of k. (Enter your answer as a comma-separated list.) λ = k = When k = -11, determine the type of the equilibrium point at the origin. The origin is a(n) ? Find the critical value or values of k at which the qualitative nature of the phase portrait changes. (If there is more than one such value, enter a comma-separated list.)
Consider the following system: 22-(2) * 7 k dt Find the eigenvalues in terms of k. (Enter your answer as a comma-separated list.) λ = k = When k = -11, determine the type of the equilibrium point at the origin. The origin is a(n) ? Find the critical value or values of k at which the qualitative nature of the phase portrait changes. (If there is more than one such value, enter a comma-separated list.)
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
![### System of Differential Equations
Consider the following system:
\[
\frac{d}{dt} \vec{x} = \begin{pmatrix}
0 & -9 \\
7 & k
\end{pmatrix} \vec{x}
\]
1. **Find the eigenvalues in terms of \( k \). (Enter your answer as a comma-separated list.)**
\[
\lambda = \_\_\_
\]
2. **When \( k = -11 \), determine the type of equilibrium point at the origin.**
\[
\text{The origin is a(n) } \_\_\_
\]
3. **Find the critical value or values of \( k \) at which the qualitative nature of the phase portrait changes. (If there is more than one such value, enter a comma-separated list.)**
\[
k = \_\_\_
\]
4. **Draw phase portraits for values of \( k \) slightly above and slightly below each of the critical values you found.**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F216015d9-daff-4e98-b3d1-7f3f5202e820%2F1ebd1f9d-54ec-46c2-8444-8bf0e222d8ce%2Frdkv0j9_processed.png&w=3840&q=75)
Transcribed Image Text:### System of Differential Equations
Consider the following system:
\[
\frac{d}{dt} \vec{x} = \begin{pmatrix}
0 & -9 \\
7 & k
\end{pmatrix} \vec{x}
\]
1. **Find the eigenvalues in terms of \( k \). (Enter your answer as a comma-separated list.)**
\[
\lambda = \_\_\_
\]
2. **When \( k = -11 \), determine the type of equilibrium point at the origin.**
\[
\text{The origin is a(n) } \_\_\_
\]
3. **Find the critical value or values of \( k \) at which the qualitative nature of the phase portrait changes. (If there is more than one such value, enter a comma-separated list.)**
\[
k = \_\_\_
\]
4. **Draw phase portraits for values of \( k \) slightly above and slightly below each of the critical values you found.**
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