Problem 6 - Calculus

Question

Find all equilibrium points. Give answers as ordered pairs \((x, y).\) $$\begin{aligned} &\frac{d x}{d t}=-2 x+4 x y\\\ &\frac{d y}{d t}=-8 y+2 x y \end{aligned}$$

Step-by-Step Solution

Answer

The equilibrium points are (0, 0) and (4, 0.5).

Step 1: Understanding Equilibrium Points

Equilibrium points occur where both derivatives \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \) are equal to zero. This represents a point where the system is in a steady state, and neither \( x \) nor \( y \) is changing with respect to time.

Step 2: Set the Derivatives to Zero

Set \( \frac{dx}{dt} = -2x + 4xy = 0 \) and \( \frac{dy}{dt} = -8y + 2xy = 0 \). This will give us two equations to solve simultaneously.

Step 3: Solve for Common Terms

From \( \frac{dx}{dt} = 0 \):\[-2x + 4xy = 0 \Rightarrow x(-2 + 4y) = 0.\] Thus, either \( x = 0 \) or \( 4y = 2 \) which simplifies to \( y = \frac{1}{2} \).

Step 4: Solve the Second Equation

Substitute the solutions found from \( x(-2 + 4y) = 0 \) into the second equation:\[-8y + 2xy = 0.\]For \( x = 0 \): \[-8y = 0 \Rightarrow y = 0.\]For \( y = \frac{1}{2} \), substitute into the second equation: \[-8\left(\frac{1}{2}\right) + 2x\left(\frac{1}{2}\right) = 0 \Rightarrow -4 + x = 0 \Rightarrow x = 4.\]

Step 5: List the Equilibrium Points

From our solutions, we have two equilibrium points:1. \((x, y) = (0, 0)\) when both variables are zero.2. \((x, y) = (4, \frac{1}{2})\) from substituting values into the simplified equations.

Key Concepts

Differential Equations

Differential equations are mathematical equations that involve the rates of change of quantities. In the context of this problem, we have two differential equations that describe how the quantities \( x \) and \( y \) change over time:
\[\begin{aligned} \frac{d x}{d t} &= -2 x + 4 x y,\ \frac{d y}{d t} &= -8 y + 2 x y. \end{aligned}\]
These equations are essential for modeling real-world phenomena where the rate of change is a crucial factor.

Key features of differential equations include:

• They describe how a quantity changes over time.
• Their solutions are functions, such as \( x(t) \) and \( y(t) \), representing the change of \( x \) and \( y \) as time progresses.
• They can be either ordinary differential equations (ODEs) when involving single independent variables or partial differential equations (PDEs) with multiple variables.

In systems like this, each equation is often dependent on multiple variables and their interactions, making the analysis of equilibrium points crucial to understanding the system's long-term behavior.

Systems of Equations

A system of equations consists of a set of equations with multiple unknowns. In this exercise, our system comprises two equations, each involving \( x \) and \( y \):
\[\begin{aligned} \frac{d x}{d t} &= -2 x + 4 x y,\ \frac{d y}{d t} &= -8 y + 2 x y. \end{aligned}\]
Here, we aim to find solutions that satisfy both equations simultaneously.

To explore equilibrium points, we make both equations zero because that indicates no change, or equilibrium.

• This involves solving for \( x \) and \( y \) when \( \frac{dx}{dt} = 0 \) and \( \frac{dy}{dt} = 0 \).
• We obtain the system:
  \[ \begin{aligned} -2x + 4xy &= 0,\ -8y + 2xy &= 0. \end{aligned} \]
• By setting common terms to zero, we simplify and solve the equations, leading us to equilibrium points.

Understanding systems of equations is crucial for dissecting how variables interact and determine the system's behavior at various states.

Steady State Analysis

Steady state analysis is about determining when a system reaches an unchanging condition, known as equilibrium. In such states, the system remains constant over time, which is invaluable for predicting long-term behavior.

This problem involves finding equilibrium points for a dynamic system, expressed through differential equations. These points are significant because they represent the conditions under which the system stops evolving.

To find these, we:

• Set each equation's derivative to zero: \( \frac{dx}{dt} = 0 \) and \( \frac{dy}{dt} = 0 \).
• Solve these conditions to find values of \( x \) and \( y \) - the equilibrium points.

For this exercise, equilibrium points are:

• \((0,0)\): the system is at rest with both variables at zero.
• \((4, \frac{1}{2})\): a non-trivial equilibrium where the variables balance perfectly.

Such analyses are pivotal for understanding complex systems, as they provide insights into the conditions under which stability occurs, allowing for better control or prediction of outcomes.