a. Find the most general real-valued solution to the linear system of differential equations a' = x₁(t) x₂ (t) ] C1 b. In the phase plane, this system is best described as a source/ unstable node sink / stable node saddle O center point / ellipses spiral source spiral sink none of these + C₂ 15-16 12 -13 x.
a. Find the most general real-valued solution to the linear system of differential equations a' = x₁(t) x₂ (t) ] C1 b. In the phase plane, this system is best described as a source/ unstable node sink / stable node saddle O center point / ellipses spiral source spiral sink none of these + C₂ 15-16 12 -13 x.
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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![### Transcription for Educational Website
---
**Problem:**
a. Find the most general real-valued solution to the linear system of differential equations \(\mathbf{\dot{x}} = \begin{bmatrix} 15 & -16 \\ 12 & -13 \end{bmatrix} \mathbf{x}\).
\[
\begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix} = c_1 \begin{bmatrix} \text{\_\_\_\_} \\ \text{\_\_\_\_} \end{bmatrix} + c_2 \begin{bmatrix} \text{\_\_\_\_} \\ \text{\_\_\_\_} \end{bmatrix}
\]
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
---
**Explanation:**
- **Part (a):** The problem requires finding the general solution of a linear system of differential equations, represented by a matrix. The solution will be expressed as a linear combination of vectors using constants \(c_1\) and \(c_2\).
- **Part (b):** The system's behavior in the phase plane should be categorized. Options include different types of equilibrium points such as sources, sinks, nodes, and more, based on the eigenvalues and eigenvectors of the matrix provided.
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Transcribed Image Text:### Transcription for Educational Website
---
**Problem:**
a. Find the most general real-valued solution to the linear system of differential equations \(\mathbf{\dot{x}} = \begin{bmatrix} 15 & -16 \\ 12 & -13 \end{bmatrix} \mathbf{x}\).
\[
\begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix} = c_1 \begin{bmatrix} \text{\_\_\_\_} \\ \text{\_\_\_\_} \end{bmatrix} + c_2 \begin{bmatrix} \text{\_\_\_\_} \\ \text{\_\_\_\_} \end{bmatrix}
\]
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
---
**Explanation:**
- **Part (a):** The problem requires finding the general solution of a linear system of differential equations, represented by a matrix. The solution will be expressed as a linear combination of vectors using constants \(c_1\) and \(c_2\).
- **Part (b):** The system's behavior in the phase plane should be categorized. Options include different types of equilibrium points such as sources, sinks, nodes, and more, based on the eigenvalues and eigenvectors of the matrix provided.
---
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