Problem 5 - Calculus

Question

Find all equilibrium points. Give answers as ordered pairs \((x, y).\) $$\begin{aligned} &\frac{d x}{d t}=-3 x+x y\\\ &\frac{d y}{d t}=5 y-x y \end{aligned}$$

Step-by-Step Solution

Answer

The equilibrium points are (0, 0) and (5, 3).

Step 1: Define Equilibrium Points

Equilibrium points occur where the derivatives are equal to zero. For this system, this means solving the two equations: \( \frac{dx}{dt} = -3x + xy = 0 \) and \( \frac{dy}{dt} = 5y - xy = 0 \).

Step 2: Solve the First Equation

The first equation is \( -3x + xy = 0 \). Factor out \( x \) to get \( x(-3 + y) = 0 \). This implies two cases: \( x = 0 \) or \( y = 3 \).

Step 3: Solve the Second Equation

The second equation is \( 5y - xy = 0 \). Factor out \( y \) to get \( y(5 - x) = 0 \). This implies two cases: \( y = 0 \) or \( x = 5 \).

Step 4: Combine Results from Equations

Now we combine solutions from both equations: - If \( x = 0 \), then from the second equation, either \( y = 0 \) or \( x = 5 \) (which doesn't add any condition since \( x = 0 \) in this assumption). Hence, \( (0, 0) \) is an equilibrium point.- If \( y = 3 \), from the second equation, \( 3(5 - x) = 0 \) which gives \( x = 5 \). So \( (5, 3) \) is an equilibrium point.- If \( y = 0 \), then from the first equation either \( x = 0 \) or \( y = 3 \) (which doesn't add any due to \( y = 0 \) in this assumption). Hence, \( (0, 0) \) is already considered.

Key Concepts

Differential Equations

Differential equations are mathematical expressions that describe rates of change. These equations are fundamental in capturing how systems evolve over time.
They are used in various fields such as physics, engineering, biology, and economics.
In this context, we are dealing with a system of two differential equations:

• \( \frac{d x}{d t} = -3x + xy \)
• \( \frac{d y}{d t} = 5y - xy \)

Each equation illustrates how one variable depends on another. The first equation models the change in \( x \) with respect to time and is influenced by both \( x \) and \( y \).
The second equation shows how \( y \) changes with time, also depending on \( x \) and \( y \).
Understanding these relationships helps in analyzing how the system's variables interact and balance each other over time.

Phase Plane Analysis

Phase plane analysis is a graphical method to study the behavior of systems of differential equations. It involves plotting the trajectories of the system in a plane where the axes represent different variables, such as \( x \) and \( y \) in our case.
This method provides a visual overview of how the system evolves.
Points on this plane are called state points and they represent states of the system at particular times. As the system evolves, these points move, tracing out curves known as trajectories.

• Trajectories convey how one state transitions into another.
• Equilibrium points, also called fixed points, are where the system remains constant over time.

In our exercise, we find the equilibrium points, meaning where the curves intersect at a standstill. These are given as the solutions \((0, 0)\) and \((5, 3)\) from the equations.
By applying phase plane analysis, we can better understand the stability and overall dynamics of the system.

Stability Analysis

Stability analysis involves examining equilibrium points to determine if these points are stable or unstable. In simple terms, we want to understand what happens when the system is disturbed from these points.
Stable equilibrium means that if the system experiences a small disturbance, it will return to the equilibrium point.

Unstable equilibrium implies that a slight disturbance can lead the system away from the equilibrium, resulting in new behavior patterns.

In our exercise, after finding two equilibrium points \((0, 0)\) and \((5, 3)\), the next step involves examining the nature of these points.
Using mathematical criteria such as the Jacobian matrix and eigenvalues, one can categorize the equilibrium as:

• Stable (attracting spiral or node)
• Unstable (repelling spiral or node)
• Semistable (a saddle point)

This analysis is crucial because it provides insights into the long-term behavior of the system. Understanding stability helps predict if systems will settle into predictable patterns or exhibit chaotic motion over time.