Question

In Exercises \(5-7\), find the eigenvalues and eigenvectors with the eig and null commands, as demonstrated in Example 4 of Chapter 12. You may find format rat helpful. Then enter the system into pplane6, and draw the straight line solutions. For example, if one eigenvector happens to be \(\mathbf{v}=[1,-2]^{T}\), use the Keyboard input window to start straight line solutions at \((1,-2)\) and \((-1,2)\). Perform a similar task for the other eigenvector. Finally, the straight line solutions in these exercises divide the phase plane into four regions. Use your mouse to start several solution trajectories in each region. $$ \begin{aligned} &x^{\prime}=6 x-y \\ &y^{\prime}=-3 y \end{aligned} $$

Short answer

Eigenvalues: 6, -3; eigenvectors: \( \begin{pmatrix} 1 \\ 0 \end{pmatrix} \), \( \begin{pmatrix} 1 \\ 9 \end{pmatrix} \).

Step-by-step solution

Write Down the System

We are given the linear system: \[ x' = 6x - y, \ y' = -3y. \] Our task is to analyze this system to find eigenvalues and eigenvectors, use pplane6 for visualization, and start solution trajectories as instructed.

Write the System in Matrix Form

The system can be expressed in a matrix form as:\[ \begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} 6 & -1 \ 0 & -3 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} \]Here, the matrix \( A \) is \( \begin{pmatrix} 6 & -1 \ 0 & -3 \end{pmatrix} \).

Find Eigenvalues

To find the eigenvalues \( \lambda \) of the matrix \( A \), solve the characteristic equation:\[ \det(A - \lambda I) = 0 \]For our matrix \( A \):\[ \begin{vmatrix} 6 - \lambda & -1 \ 0 & -3 - \lambda \end{vmatrix} = 0 \]This simplifies to:\[(6 - \lambda)(-3 - \lambda) - 0 = 0\]Solving gives eigenvalues \( \lambda_1 = 6 \) and \( \lambda_2 = -3 \).

Find Eigenvectors

For each eigenvalue, find the corresponding eigenvector.- **Eigenvector for \( \lambda_1 = 6 \):** Solve \( (A - 6I)\mathbf{v} = 0 \): \[ \begin{pmatrix} 0 & -1 \ 0 & -9 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 0 \ 0 \end{pmatrix} \] Simplifying gives \( y = 0 \), and any \( x \) works, so \( \mathbf{v}_1 = \begin{pmatrix} 1 \ 0 \end{pmatrix} \).- **Eigenvector for \( \lambda_2 = -3 \):** Solve \( (A + 3I)\mathbf{v} = 0 \): \[ \begin{pmatrix} 9 & -1 \ 0 & 0 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 0 \ 0 \end{pmatrix} \] Simplifying gives \( x = \frac{1}{9}y \), thus one eigenvector is \( \mathbf{v}_2 = \begin{pmatrix} 1 \ 9 \end{pmatrix} \).

Visualize with pplane6

Use pplane6 to enter the system of equations and visualize the phase plane.- Enter the system equations directly.- Use the eigenvectors to start straight-line solutions: - For \( \mathbf{v}_1 = \begin{pmatrix} 1 \ 0 \end{pmatrix} \), start solutions at \((1, 0)\) and \((-1, 0)\). - For \( \mathbf{v}_2 = \begin{pmatrix} 1 \ 9 \end{pmatrix} \), start solutions at \((1, 9)\) and \((-1, -9)\).- The straight-line solutions divide the phase plane into four regions. Start trajectories in each region using the mouse.

Key concepts

Linear System

A linear system refers to a set of linear equations that can describe various complex processes, such as dynamic systems in mathematics and engineering. In this exercise, the given system is \[ x' = 6x - y, \ y' = -3y. \] Each equation represents how one variable affects another over time, a common characteristic of dynamic systems.
To understand these interactions, solving the linear system involves several analytical methods, including representing it in a matrix form and finding eigenvalues and eigenvectors. This helps to predict system behavior and interactions between variables. Linear systems are foundational in understanding complex systems in physics, biology, and economics.
By exploring the behavior of these systems in various initial conditions, they can demonstrate behaviors such as growth, decay, or stabilization.

Phase Plane Analysis

Phase plane analysis is a graphical method to study differential equations, focusing on visualizing the trajectories of systems of first-order equations. It's particularly useful for understanding the qualitative behavior of dynamical systems over time.

• The phase plane is a 2-dimensional plot where each point represents a state of the system, defined by the values of variables, such as \(x\) and \(y\) in our system.
• Trajectories in the phase plane show how the system evolves from different initial states.

In this exercise, the goal of phase plane analysis is to identify straight-line solutions that divide the plane into distinct regions, each indicating a different type of system behavior. By plotting these trajectories, we can predict the system's future behaviors without solving the system analytically, offering insight into stability and response to changes.

Characteristic Equation

The characteristic equation is essential for finding eigenvalues of a matrix, which are critical in understanding system behavior. The characteristic equation is derived from the matrix equation \( \det(A - \lambda I) = 0 \), where \(A\) is the system matrix and \(\lambda\) represents the eigenvalues.
For a given linear system, the eigenvalues tell us how solutions grow or decay over time, showing rates at which these changes happen.

• In the current problem, after calculating the determinant of \( A - \lambda I \), we solve to find \( \lambda_1 = 6 \) and \( \lambda_2 = -3 \).

Understanding these values is crucial for predicting system behavior.
Positive eigenvalues typically indicate growth or instability, while negative ones suggest decay or stabilization of certain states.

Matrix Form

The matrix form of a system of equations is a compact and powerful representation that simplifies the analysis of linear systems. The matrix form transforms the given system into matrix multiplication:
\[ \begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} 6 & -1 \ 0 & -3 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix}. \]
This representation aligns with linear algebra techniques, making it easier to find eigenvalues and eigenvectors, solve systems, and understand their properties.

• The matrix, often denoted as \(A\), encodes the system's dynamics into a single entity, allowing compact operations on the system.
• Matrix manipulation reveals insights into the system's behavior, enabling analysis that is often less intuitive in the standard equation form.

Using a matrix form lays the groundwork for computational tools to further analyze and visualize the system behaviors.

Visualization with pplane6

Pplane6 is a MATLAB tool used for the phase plane analysis of dynamical systems. It offers a hands-on method for visualizing how systems behave under different conditions by plotting their trajectories.

• With pplane6, users can input differential equations and observe how solutions evolve over time by plotting in the phase plane.
• It allows for starting solution trajectories based directly on eigenvectors and provides a graphic means to identify key behavioral aspects, like dividing lines and regions of attraction.

This makes it an ideal tool for exploring systems like the one in this exercise, allowing students to observe in real-time how theory translates into practice.
The interactive nature of pplane6 helps deepen understanding of dynamic systems by bridging the gap between abstract equations and observable behaviors.