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Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Problem Statement:**
a. Find the most general real-valued solution to the linear system of differential equations \(\vec{x}' = \begin{bmatrix} -10 & -9 \\ 6 & 5 \end{bmatrix} \vec{x}\).
**Solution:**
The general solution to the system is given by:
\[
\begin{bmatrix}
x_1(t) \\
x_2(t)
\end{bmatrix}
=
c_1
\begin{bmatrix}
1 \\
1
\end{bmatrix}
+
c_2
\begin{bmatrix}
\frac{3}{2} \\
1
\end{bmatrix}
\]
**Explanation:**
The solution involves finding the eigenvectors corresponding to the eigenvalues of the matrix, then expressing the solution as a linear combination of these eigenvectors.
b. In the phase plane, this system is best described as a:
- ○ source / unstable node
- ● sink / stable node
- ○ saddle
- ○ center point / ellipses
- ○ spiral source
- ○ spiral sink
- ○ none of these
**Selection:**
- The system is best described as a **sink / stable node** (indicated by the blue circle). This means trajectories move towards the fixed point, indicating stability in the phase plane.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2eee5fef-24c0-42d2-a99f-ec7ca4df2e97%2F54035295-aec3-4caa-af82-e9a1e08ba685%2F6peut8_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
a. Find the most general real-valued solution to the linear system of differential equations \(\vec{x}' = \begin{bmatrix} -10 & -9 \\ 6 & 5 \end{bmatrix} \vec{x}\).
**Solution:**
The general solution to the system is given by:
\[
\begin{bmatrix}
x_1(t) \\
x_2(t)
\end{bmatrix}
=
c_1
\begin{bmatrix}
1 \\
1
\end{bmatrix}
+
c_2
\begin{bmatrix}
\frac{3}{2} \\
1
\end{bmatrix}
\]
**Explanation:**
The solution involves finding the eigenvectors corresponding to the eigenvalues of the matrix, then expressing the solution as a linear combination of these eigenvectors.
b. In the phase plane, this system is best described as a:
- ○ source / unstable node
- ● sink / stable node
- ○ saddle
- ○ center point / ellipses
- ○ spiral source
- ○ spiral sink
- ○ none of these
**Selection:**
- The system is best described as a **sink / stable node** (indicated by the blue circle). This means trajectories move towards the fixed point, indicating stability in the phase plane.
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