Consider the following system: k 2²-(33) ² = dt k= Find the eigenvalues in terms of k. (Enter your answer as a comma-separated list.) λ = When k = 4, determine the type of the equilibrium point at the origin. The origin is a(n) ? k 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.) Draw phase portraits for values of k slightly above and slightly below each of the critical values you found.
Consider the following system: k 2²-(33) ² = dt k= Find the eigenvalues in terms of k. (Enter your answer as a comma-separated list.) λ = When k = 4, determine the type of the equilibrium point at the origin. The origin is a(n) ? k 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.) Draw phase portraits for values of k slightly above and slightly below each of the critical values you found.
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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![### Analyzing the Given System of Differential Equations
Consider the following system of differential equations:
\[
\frac{d}{dt} \vec{x} = \begin{pmatrix} k & 3 \\ -3 & k \end{pmatrix} \vec{x}
\]
#### Tasks:
1. **Find the eigenvalues in terms of \( k \).**
(Enter your answer as a comma-separated list.)
**Eigenvalues:**
\[
\lambda = \ \_\_\_\_
\]
2. **Determine the type of the equilibrium point at the origin when \( k = 4 \).**
**The origin is a(n):**
\[
\boxed{\ \_\_\_\_ \ }
\]
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.)
**Critical values:**
\[
k = \ \_\_\_\_
\]
4. **Draw phase portraits for values of \( k \) slightly above and slightly below each of the critical values you found.**
### Explanation of Tacks:
1. **Eigenvalues in terms of \( k \):**
- To find the eigenvalues, solve the characteristic equation derived from the determinant:
\[
\text{det}\left( \begin{pmatrix} k & 3 \\ -3 & k \end{pmatrix} - \lambda \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \right) = 0
\]
2. **Equilibrium Point when \( k = 4 \):**
- Substitute \( k = 4 \) into the system and analyze the nature of the equilibrium point by examining the sign and discriminant of the eigenvalues.
3. **Critical Values of \( k \):**
- Find the values of \( k \) at which the nature of the eigenvalues changes (e.g., from real to complex, or vice versa), which indicates a change in the qualitative nature of the phase portrait of the system.
4. **Phase Portraits:**
- Based on the identified critical values, sketch phase portraits for values of \( k \) slightly above and below each critical value](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F216015d9-daff-4e98-b3d1-7f3f5202e820%2F0603d894-2b73-41f3-b2b7-89574cd6e2c6%2Fuzn1bwv_processed.png&w=3840&q=75)
Transcribed Image Text:### Analyzing the Given System of Differential Equations
Consider the following system of differential equations:
\[
\frac{d}{dt} \vec{x} = \begin{pmatrix} k & 3 \\ -3 & k \end{pmatrix} \vec{x}
\]
#### Tasks:
1. **Find the eigenvalues in terms of \( k \).**
(Enter your answer as a comma-separated list.)
**Eigenvalues:**
\[
\lambda = \ \_\_\_\_
\]
2. **Determine the type of the equilibrium point at the origin when \( k = 4 \).**
**The origin is a(n):**
\[
\boxed{\ \_\_\_\_ \ }
\]
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.)
**Critical values:**
\[
k = \ \_\_\_\_
\]
4. **Draw phase portraits for values of \( k \) slightly above and slightly below each of the critical values you found.**
### Explanation of Tacks:
1. **Eigenvalues in terms of \( k \):**
- To find the eigenvalues, solve the characteristic equation derived from the determinant:
\[
\text{det}\left( \begin{pmatrix} k & 3 \\ -3 & k \end{pmatrix} - \lambda \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \right) = 0
\]
2. **Equilibrium Point when \( k = 4 \):**
- Substitute \( k = 4 \) into the system and analyze the nature of the equilibrium point by examining the sign and discriminant of the eigenvalues.
3. **Critical Values of \( k \):**
- Find the values of \( k \) at which the nature of the eigenvalues changes (e.g., from real to complex, or vice versa), which indicates a change in the qualitative nature of the phase portrait of the system.
4. **Phase Portraits:**
- Based on the identified critical values, sketch phase portraits for values of \( k \) slightly above and below each critical value
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