### Analyzing Linear Systems with Eigenvalues and Eigenvectors**Problem Overview**Each of the linear systems below has one eigenvalue and one line of eigenvectors. For each system, you are asked to:(a) Find the eigenvalue;(b) Find an eigenvector;(c) Sketch the direction field;(d) Sketch the phase portrait, including the solution curve with the initial condition $ Y_0 = \begin{pmatrix} 1 \\ 0 \end{pmatrix} $;(e) Sketch the $ x(t) $- and $ y(t) $-graphs of the solution with the initial condition $ Y_0 = \begin{pmatrix} 1 \\ 0 \end{pmatrix} $.**Given Linear System**\[ \frac{dY}{dt} = \begin{pmatrix} 0 & 1 \\ -1 & -2 \end{pmatrix} Y \]### Steps to Solve1. **Find the Eigenvalue** Determine the eigenvalue ($\lambda$) by solving the characteristic equation derived from the matrix: \[ \det(A - \lambda I) = 0 \] where \[ A = \begin{pmatrix} 0 & 1 \\ -1 & -2 \end{pmatrix} \] 2. **Find an Eigenvector** For the eigenvalue found in step 1, determine an associated eigenvector ($v$) by solving: \[ (A - \lambda I)v = 0 \]3. **Sketch the Direction Field** Visualize the direction field by plotting the vector field defined by the linear system.4. **Sketch the Phase Portrait** Include the solution curve for the initial condition $ Y_0 = \begin{pmatrix} 1 \\ 0 \end{pmatrix} $. The phase portrait displays the trajectories of the system in the phase plane.5. **Sketch the $ x(t) $- and $ y(t) $-graphs** Plot the time-dependent solutions $x(t)$ and $y(t)$ by solving the system of differential equations with the given initial condition.**Description of Graphs and Diagrams**- **Direction Field**: A plot of vectors representing the direction and magnitude of the rate of change of $ Y $ at various points in the phase space.

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Each of the linear systems has one eigenvalue and one line of eigenvectors. For each system, (a) find the eigenvalue; (b) find an eigenvector; (9 sketch the direction field;

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

### Analyzing Linear Systems with Eigenvalues and Eigenvectors

**Problem Overview**

Each of the linear systems below has one eigenvalue and one line of eigenvectors. For each system, you are asked to:

(a) Find the eigenvalue;
(b) Find an eigenvector;
(c) Sketch the direction field;
(d) Sketch the phase portrait, including the solution curve with the initial condition $ Y_0 = \begin{pmatrix} 1 \\ 0 \end{pmatrix} $;
(e) Sketch the $ x(t) $- and $ y(t) $-graphs of the solution with the initial condition $ Y_0 = \begin{pmatrix} 1 \\ 0 \end{pmatrix} $.

**Given Linear System**

\[ \frac{dY}{dt} = \begin{pmatrix} 0 & 1 \\ -1 & -2 \end{pmatrix} Y \]

### Steps to Solve

1. **Find the Eigenvalue**
Determine the eigenvalue ($\lambda$) by solving the characteristic equation derived from the matrix:
\[ \det(A - \lambda I) = 0 \]
where
\[ A = \begin{pmatrix} 0 & 1 \\ -1 & -2 \end{pmatrix} \]

2. **Find an Eigenvector**
For the eigenvalue found in step 1, determine an associated eigenvector ($v$) by solving:
\[ (A - \lambda I)v = 0 \]

3. **Sketch the Direction Field**
Visualize the direction field by plotting the vector field defined by the linear system.

4. **Sketch the Phase Portrait**
Include the solution curve for the initial condition $ Y_0 = \begin{pmatrix} 1 \\ 0 \end{pmatrix} $. The phase portrait displays the trajectories of the system in the phase plane.

5. **Sketch the $ x(t) $- and $ y(t) $-graphs**
Plot the time-dependent solutions $x(t)$ and $y(t)$ by solving the system of differential equations with the given initial condition.

**Description of Graphs and Diagrams**

- **Direction Field**: A plot of vectors representing the direction and magnitude of the rate of change of $ Y $ at various points in the phase space.

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