Question

(a) Find all the critical points (equilibrium solutions). (b) Use a computer to draw a direction field and portrait for the system. (c) From the plot(s) in part (b) determine whether each critical point is asymptotically stable, stable, or unstable, and classify it as to type. $$ d x / d t=(2+x)(y-x), \quad d y / d t=(4-x)(y+x) $$

Short answer

#Answer# The critical points are (-2, 2), (0, 0), and (4, 4). After analyzing the direction field and portrait of the system, classify the stability and type of each critical point as follows: 1. Critical point (-2, 2): If the trajectories are moving towards this point, it is asymptotically stable and can be classified as an attracting node or spiral. 2. Critical point (0, 0): If the trajectories are neither moving towards nor away from this point, it is stable, and can be classified as a saddle point. 3. Critical point (4, 4): If the trajectories are moving away from this point, it is unstable and can be classified as a repelling node or spiral. (Note: The specific type and stability of each critical point will depend on the direction field and portrait results, which are not provided here).

Step-by-step solution

Find the critical points (equilibrium solutions)

To find the critical points, we set the right sides of both differential equations to zero and solve for x and y: $$ (2+x)(y-x) = 0, \quad (4-x)(y+x) = 0 $$ We have two possible cases: (i) Either \(2+x=0\) or \(y-x=0\) (ii) Either \(4-x=0\) or \(y+x=0\)

Step 1.1: Solve for x and y in case (i)

From the first case we have: (i) Either \(x=-2\) or \(x=y\) Now we substitute these values in the second equation: - If \(x=-2\): \((4-(-2))(y+(-2)) = 6(y-2) = 0 \Rightarrow y=2\) - If \(x=y\): \((4-y)(2y) = 0 \Rightarrow y=0\) or \(y=4\) In case (i), we found critical points as \((-2, 2)\), \((0, 0)\) and \((4, 4)\).

Step 1.2: Solve for x and y in case (ii)

From the second case we have: (ii) Either \(4-x=0\) or \(y+x=0\) Now we substitute these values in the first equation: - If \(x=4\): \((2+4)(y-4) = 6(y-4)=0 \Rightarrow y=4\) - If \(x=-y\): \((2-y)(2y)=0 \Rightarrow y=0\) or \(y=2\). But if \(y=2\), then \(x=-2\), which was already considered in case (i). In case (ii), we found critical points as \((4, 4)\) and \((0, 0)\).

Direction field and portrait (use a computer)

Using a computer software like Python or MATLAB, plot the direction field and portrait for the given system of differential equations.

Analyze the plot and determine stability of the critical points

By examining the direction field and portrait of the system, classify the stability and type of each critical point. 1. If the trajectories are moving towards the critical point, it is asymptotically stable. 2. If the trajectories are neither moving towards nor away from the critical point, it is stable. 3. If the trajectories are moving away from the critical point, it is unstable. Types of critical points include: - Nodes (attracting, repelling, or saddle) - Spirals (attracting or repelling) Using the stability analysis and classification of the critical points from the plot, determine the stability and type for each critical point found in step 1.

Key concepts

Critical Points

In the study of differential equations, critical points, also known as equilibrium solutions, are vital for understanding the behavior of dynamic systems. These are the points where the system's variables do not change over time. To find these points, we set the derivatives equal to zero and solve the resulting equations. In our example, this involves solving the system:

• \((2+x)(y-x) = 0\)
• \((4-x)(y+x) = 0\)

By examining each case where either factor in each equation is zero, we discover the critical points: \((-2, 2)\), \(0, 0\), and \(4, 4\). These points are where the system balances itself, and knowing them helps us predict the future behavior of the system.

Stability Analysis

Stability analysis involves examining how a system behaves near its critical points. Specifically, whether the system returns to its equilibrium state or diverges away from it after a slight disturbance. In simple terms, stability tells us whether a point is:

• Asymptotically stable, where trajectories converge to the point.
• Stable, where trajectories neither attract nor repel.
• Unstable, where trajectories diverge from the point.

In our exercise, after identifying the critical points, we evaluate their stability using a direction field plot. This visual tool illuminates the behavior of each nearby trajectory, offering insights into whether the system will settle, oscillate, or move away from these points.

Equilibrium Solutions

Equilibrium solutions are at the heart of many real-world systems, indicating states where the system's variables do not change over time. Such solutions offer vital insights into how systems naturally settle, balance, or evolve into stable forms. Once critical points are found, they are examined to determine if slight changes in the system return to equilibrium, stay the same, or diverge.
The equilibrium solutions, in our problem, found at \((-2, 2)\), \(0, 0\), and \(4, 4\), showcase different behaviors and stabilities that offer a glimpse into the overall system dynamics. Comprehending these points ensures that we can predict the system’s response to various initial conditions.

Direction Fields

A direction field, sometimes referred to as a slope field, is a graphical representation of a differential equation. It shows the trends and tendencies of the system's behavior without having to solve the equations explicitly. By plotting small line segments at various points, it provides an immediate visual clue to the trajectory of the system's solutions.
When approaching problems with differential equations, viewing the direction field can significantly aid in visualizing how solutions evolve over time. It captures the essence of the system's dynamics, offering a snapshot of how trajectories behave, converge, or diverge in relation to critical points. In our exercise, this involves using computer-generated plots to understand how each critical point behaves, thereby determining their stability classifications.