Problem 11 - Advanced Engineering Mathematics

Question

In Problems, find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=x\left(10-x-\frac{1}{2} y\right) \\ &y^{\prime}=y(16-y-x) \end{aligned} $$

Step-by-Step Solution

Answer

Critical points are \( (0, 16) \), \( (10, 0) \), and \( (4, 12) \).

Step 1: Understand the given system

Before finding the critical points, we need to understand what the system of differential equations represents. The given system is \( x' = x(10 - x - \frac{1}{2}y) \) and \( y' = y(16 - y - x) \). These equations describe how the values of \( x \) and \( y \) change over time.

Step 2: Set up the equations for critical points

Critical points occur where both derivatives are zero. This means we need to solve the equations \( x' = 0 \) and \( y' = 0 \) simultaneously. We will solve each equation for zero to find the critical points.

Step 3: Solve for \( x' = 0 \)

The equation \( x' = 0 \) simplifies to \( x(10 - x - \frac{1}{2}y) = 0 \). This can give us two potential conditions: \( x = 0 \) or \( 10 - x - \frac{1}{2}y = 0 \).

Step 4: Solve for \( y' = 0 \)

The equation \( y' = 0 \) simplifies to \( y(16 - y - x) = 0 \). This can also give us two potential conditions: \( y = 0 \) or \( 16 - y - x = 0 \).

Step 5: Find intersections of conditions

To find the critical points, we need to find where the solutions from the \( x' = 0 \) and \( y' = 0 \) equations intersect.1. If \( x = 0 \): - From \( 16 - y - x = 0 \), we have \( 16 - y = 0 \) ➔ \( y = 16 \). - Critical point: \( (0, 16) \).2. If \( y = 0 \): - From \( 10 - x - \frac{1}{2}y = 0 \), we simplify to \( 10 - x = 0 \) ➔ \( x = 10 \). - Critical point: \( (10, 0) \).3. From both non-zero conditions: - \( 10 - x - \frac{1}{2}y = 0 \) ➔ \( x = 10 - \frac{1}{2}y \) - \( 16 - y - x = 0 \), substitute \( x = 10 - \frac{1}{2}y \). - Solve \( 16 - y - (10 - \frac{1}{2}y) = 0 \) ➔ \( 16 - y - 10 + \frac{1}{2}y = 0 \), - \( 6 = \frac{1}{2}y \implies y = 12 \). - Substitute back: \( x = 10 - \frac{1}{2}(12) = 4 \). - Critical point: \( (4, 12) \).

Step 6: List all critical points found

From our analysis, the critical points of the system are \( (0, 16) \), \( (10, 0) \), and \( (4, 12) \). These are the points where the system has no change in \( x \) and \( y \) over time.

Key Concepts

Critical Points

In the context of autonomous systems, **critical points** are those magical places where the system changes its behavior. These are the points

• where the derivatives of the system are zero.
• Here, both variables don't change, acting as potential constants.
• They help understand the system's steady state or equilibrium.

To find critical points, we solve the differential equations with both derivatives equal to zero. More technically, for any system \(x' = f(x, y)\) and \(y' = g(x, y)\), set

• \(f(x, y) = 0\) and
• \(g(x, y) = 0\)

We find points that satisfy these conditions. These points differ based on the nature of each equation's components.
In our exercise, by solving these equations together, we identified three critical points: \((0, 16)\), \((10, 0)\), and \((4, 12)\).
Critical points offer insights into how the system behaves at equilibrium, presenting conditions under which state variables won't change unless disturbed.

Plane Differential Equations

**Plane differential equations** are systems consisting of two differential equations involving two variables. They are handy for analyzing systems

• across two dimensions.
• where the path of each variable can be plotted on a coordinate plane, creating a sort of motion or flow.

In such systems, each variable's change is often dependent on the other's current state.
The given exercise is a classic example, described by - \(x' = x(10-x- \frac{1}{2}y)\) and- \(y' = y(16-y-x)\).
Visualize each equation as defining a '_field_' over a plane, detailing how values evolve over time. Therefore, the entire system illustrates how points

• travel or settle across the plane,
• showcasing the interaction between its components \(x\) and \(y\).

This graphical view aids in analyzing potential outcomes and dynamic changes over time.

System of Differential Equations

A **system of differential equations** refers to multiple equations intended to capture more complex systems with multiple variables. In this exercise,

• each differential equation defines the direction and rate of change of one variable in terms of itself and the other variable.
• Such systems are autonomous if they are free from explicit time-dependencies.

How do these systems work? Their goal is to model dynamic systems that can't be precisely detailed by one equation alone. Here, we have two variables

• \( x \), representing perhaps a population or quantity, which changes as per other influencing factors \( y \),
• \( y \) itself also being dynamic, influenced by \( x \).

Each equation in this system provides its contribution towards a collective dynamic evolution, reliant upon the other. The system given - determines interaction,- predicts behavior,- and establishes points of stability (as learned in critical points).
Such systems are incredibly beneficial, often used to simulate real-world phenomena, where understanding how quantities interlink and influence one another provides substantial insight.