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In each exercise, the eigenpairs of a \((2 \times 2)\) matrix \(A\) are given where both eigenvalues are real. Consider the phase-plane solution trajectories of the linear system \(\mathbf{y}^{\prime}=A \mathbf{y}\), where $$ \mathbf{y}(t)=\left[\begin{array}{l} x(t) \\ y(t) \end{array}\right] $$ (a) Use Table \(6.2\) to classify the type and stability characteristics of the equilibrium point at \(\mathbf{y}=\mathbf{0}\). (b) Sketch the two phase-plane lines defined by the eigenvectors. If an eigenvector is \(\left[\begin{array}{l}u_{1} \\ u_{2}\end{array}\right]\), the line of interest is \(u_{2} x-u_{1} y=0\). Solution trajectories originating on such a line stay on the line; they move toward the origin as time increases if the corresponding eigenvalue is negative or away from the origin if the eigenvalue is positive. (c) Sketch appropriate direction field arrows on both lines. Use this information to sketch a representative trajectory in each of the four phase- plane regions having these lines as boundaries. Indicate the direction of motion of the solution point on each trajectory. $$ \lambda_{1}=-2, \quad \mathbf{x}_{1}=\left[\begin{array}{l} 1 \\ 0 \end{array}\right] ; \quad \lambda_{2}=-1, \quad \mathbf{x}_{2}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right] $$

Step by step solution

01

Determine the Type and Stability of the Equilibrium Point at y=0

Given eigenvalues are both real and distinct, and have 位鈧�=-2 and 位鈧�=-1. Since both eigenvalues are negative, the equilibrium point is stable and is classified as a stable node.

02

Derive Equations for Phase-Plane Lines

We are given eigenvectors x鈧� = \begin{bmatrix} 1 \\ 0 \end{bmatrix} and x鈧� = \begin{bmatrix} 1 \\ 1 \end{bmatrix}. For an eigenvector \begin{bmatrix} u鈧� \\ u鈧� \end{bmatrix}, the corresponding line is given by u鈧倄 - u鈧亂 = 0. For x鈧�, Line 1: 0 * x - 1 * y = 0 => y = 0. For x鈧�: Line 2: 1 * x - 1 * y = 0 => x - y = 0 => y = x.

03

Sketch the Phase-Plane Lines

Sketch the phase-plane lines on the coordinate plane. Line 1 is a horizontal line along the x-axis (y = 0), and Line 2 is a diagonal line with a slope of 1 (y = x).

04

Sketch Direction Field Arrows on Both Lines

Since 位鈧� = -2 and 位鈧� = -1 are both negative, the direction field arrows should point towards the origin on both lines. Place arrows pointing inwards on Line 1 (y=0) and Line 2 (y=x).

05

Sketch a Representative Trajectory in Each of the Four Phase-Plane Regions

In the phase-plane bounded by Line 1 and Line 2, we will have four regions: Upper-left, upper-right, lower-left and lower-right. In each of these regions, sketch a representative trajectory as follows: 1. Upper-left region: The trajectory starts from a point in this region and converges to the origin along the lines y=0 and y=x. 2. Upper-right region: The trajectory starts from a point in this region, approaches the x-axis (y=0) and moves towards the origin. 3. Lower-left region: The trajectory starts from a point in this region, approaches the line y=x and moves towards the origin. 4. Lower-right region: The trajectory starts from a point in this region and converges to the origin along the lines y=0 and y=x. In each trajectory, indicate the direction of motion of the solution point with arrows. Since the equilibrium point is stable, all arrows should point towards the origin.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental to understanding the behavior of linear systems like the one described by the differential equations in the exercise. When dealing with such systems, the matrix of coefficients can reveal much about the nature of its solutions by examining its eigenvalues and eigenvectors.

Eigenvalues, denoted typically as \(\lambda\), are scalars associated with a given square matrix, and they provide information about the scale of the transformation described by that matrix on particular vectors called eigenvectors. In the context of our exercise, two real eigenvalues, \(\lambda_1 = -2\) and \(\lambda_2 = -1\), were found. These negative eigenvalues immediately tell us that the associated eigenvectors shrink towards the origin when they undergo the transformation by the matrix \(\textbf{A}\).

Eigenvectors are vectors that are not rotated by the associated matrix transformation; they are only scaled by the eigenvalue. For our exercise, the eigenvectors \(\textbf{x}_1\) and \(\textbf{x}_2\) correspond to the directions along which solutions to the differential system will not rotate, reaffirming the straight-line trajectories towards the origin. This characteristic defines the natural directions of the system's flow in the phase plane.

Stability of Equilibrium Points

The concept of stability is crucial when analyzing the behavior of a system near an equilibrium point. An equilibrium point is a state of the system where no changes occur over time, often denoted in this context as \(\mathbf{y}=\mathbf{0}\).

In our exercise, the equilibrium point's stability is determined by the sign of the eigenvalues of matrix \(\textbf{A}\). Both eigenvalues being negative is indicative of a stable equilibrium point, specifically classified as a stable node. This means any solution trajectory, which may start at any point in the vicinity of the equilibrium point, will inevitably converge to this stable node as time progresses. Such a description provides significant insight into the long-term behavior of the system, as we can predict that regardless of the starting position (except in special circumstances), the system will settle at the stable equilibrium.

Direction Fields

Direction fields, or vector fields, are visual tools used to represent the solutions to a system of differential equations. They consist of arrows drawn at sample points in the phase plane that indicate the direction and rate of change for the solution passing through that point.

In the context of the exercise, the direction field arrows along the lines defined by the eigenvectors illustrate how solutions either approach or recede from the equilibrium point. Given that we have negative eigenvalues, \(\lambda_1\) and \(\lambda_2\), the direction field arrows on both eigenvector lines point toward the origin, indicating that the solution trajectories will move towards the stable node. Mapping these arrows aids in visualizing the dynamic behavior of the system and facilitates the sketching of the representative trajectories in the phase-plane regions.

Differential Equations

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They express how the rate of change of a quantity is related to the quantity itself, and they are central to modeling change in applied mathematics, physics, engineering, and other disciplines.

In our exercise, the differential equation is given in matrix form as \(\mathbf{y}'=A\mathbf{y}\), indicating how the rate of change of the vector \(\mathbf{y}(t)\) is related to the vector \(\mathbf{y}(t)\) itself via the matrix \(\textbf{A}\). The solution to this relationship is a trajectory in the phase plane that describes how the system evolves over time. By analyzing the differential equations with the help of eigenvalues and eigenvectors, we can predict the behavior and stability of the system without having to solve the equations explicitly. Instead, we look at the qualitative behavior of the solution trajectories, which is what the exercise calls for when it asks us to sketch the phase-plane and indicate the direction of the solution's movement.