Problem 26

Question

Find the critical points and phase portrait of the given autonomous first- order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the $x y$ -plane determined by the graphs of the equilibrium solutions. $$ \frac{d y}{d x}=y(2-y)(4-y) $$

Step-by-Step Solution

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Answer

Critical points: 0 (unstable), 2 (stable), 4 (unstable). Phase portrait shows arrows towards 2 between 0 and 4.

1Step 1: Identify Critical Points

To find critical points, set the differential equation to zero: $ \frac{dy}{dx} = y(2-y)(4-y) = 0 $. This gives the critical points by setting each factor equal to zero: $ y = 0 $, $ y = 2 $, and $ y = 4 $.

2Step 2: Analyze Stability of Critical Points

Examine the sign of $ \frac{dy}{dx} $ near each critical point to determine stability. - For $ y = 0 $: Choose test points. If $ y < 0 $, $ \frac{dy}{dx} < 0 $ and if $ 0 < y < 2 $, $ \frac{dy}{dx} > 0 $, so $ y = 0 $ is unstable. - For $ y = 2 $: If $ 0 < y < 2 $, $ \frac{dy}{dx} > 0 $ and if $ 2 < y < 4 $, $ \frac{dy}{dx} < 0 $, so $ y = 2 $ is stable. - For $ y = 4 $: If $ 2 < y < 4 $, $ \frac{dy}{dx} < 0 $ and if $ y > 4 $, $ \frac{dy}{dx} > 0 $, so $ y = 4 $ is unstable.

3Step 3: Sketch Solution Curves

In the $ xy $-plane, draw horizontal lines for the equilibrium solutions $ y = 0 $, $ y = 2 $, and $ y = 4 $. - Below $ y = 0 $, arrows point downward, indicating instability. - Between $ y = 0 $ and $ y = 2 $, draw arrows pointing away from $ y = 0 $ and toward $ y = 2 $. - Between $ y = 2 $ and $ y = 4 $, arrows point downward toward $ y = 2 $, showing stability. - Above $ y = 4 $, arrows point upward, indicating instability.

Key Concepts

Critical PointsStability AnalysisDifferential Equation

Critical Points

Critical points in the context of differential equations are the values of the dependent variable where the rate of change, or derivative, is zero. In simpler terms, these are points where the system is at equilibrium, meaning no change is occurring. For the given differential equation \[ \frac{d y}{d x} = y(2-y)(4-y) \], we identify critical points by setting the equation to zero: \[ y(2-y)(4-y) = 0 \]. By solving this equation, we find the critical points to be:

$ y = 0 $
$ y = 2 $
$ y = 4 $

These points are where the system can potentially remain at rest. Critical points are foundational in understanding the behavior of dynamical systems, as they serve as a reference for further analysis. Once identified, these points allow for deeper inspection through stability analysis to predict behavior around these points.

Stability Analysis

Stability analysis involves examining the behavior of a system around critical points. It tells us whether solutions near a critical point will converge to it (stable), diverge away from it (unstable), or show a mix of behaviors (semi-stable). In our example, the stability of each critical point is analyzed by checking the sign of the derivative $ \frac{d y}{d x} $.For each critical point:

At $ y = 0 $: Testing shows that solutions in the region $ y < 0 $ have negative derivatives, meaning they move further away (downwards), while those in $ 0 < y < 2 $ have positive derivatives, moving away upwards. This makes $ y = 0 $ unstable.
At $ y = 2 $: In the region $ 0 < y < 2 $, derivatives are positive, indicating an approach towards $ y = 2 $. In contrast, for $ y > 2 $ but less than 4, derivatives are negative, signaling motion towards $ y = 2 $. This means $ y = 2 $ is a stable point.
At $ y = 4 $: For $ y $ between 2 and 4, derivatives remain negative, while beyond $ y > 4 $, derivatives switch to positive. This leads solutions away from $ y = 4 $, making it unstable.

This analysis helps predict the behavior of a system around equilibrium points, crucial for understanding the entire landscape of the system's dynamics.

Differential Equation

A differential equation is a mathematical statement involving a function and its derivatives, which shows how the rates of change are connected to the function itself. It can help describe how quantities change over time or space. The given differential equation: \[ \frac{d y}{d x} = y(2-y)(4-y) \] is an example of a first-order autonomous differential equation. Here, 'first-order' means it involves the first derivative, and 'autonomous' indicates that the independent variable does not explicitly appear in the equation.To better understand this, let's break it down:

The term $ y(2-y)(4-y) $ signifies how the variable $ y $ changes (increases or decreases) depending on its current value.
By solving for $ \frac{d y}{d x} = 0 $, we identify critical points, which highlight where change ceases and the system could potentially "rest".
This type of equation's solutions can be explored graphically, often using a phase portrait that shows equilibrium solutions and direction arrows for understanding behavior around those solutions.

Understanding differential equations is key in fields like physics, engineering, economics, and biology, where processes change over time or space. They provide a powerful framework for modeling complex, real-world systems.

Problem 26

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