Problem 7 - Advanced Engineering Mathematics

Question

In Problems \(3-10\), without solving explicitly, classify the critical points of the given first-order autonomous differential equation as either asymptotically stable or unstable. All constants are assumed to be positive. $$ \frac{d x}{d t}=k(\alpha-x)(\beta-x), \alpha>\beta $$

Step-by-Step Solution

Answer

The critical point \( x = \alpha \) is asymptotically stable, and \( x = \beta \) is unstable.

Step 1: Identify Critical Points

To find the critical points, set the differential equation equal to zero: \( k(\alpha-x)(\beta-x) = 0 \). The critical points occur where each factor can be zero: \( x = \alpha \) or \( x = \beta \).

Step 2: Analyze Stability of Critical Points

Assess the stability of each critical point by determining the sign of \( \frac{dx}{dt} \) in intervals defined by the critical points \( \beta \) and \( \alpha \). For \( x < \beta \), both \( (\alpha-x) \) and \( (\beta-x) \) are positive, making \( \frac{dx}{dt} > 0 \). Thus, solutions move away from \( x = \beta \) making it unstable.

Step 3: Continue Stability Analysis for the Second Interval

For \( x = \beta \), where \( \beta < x < \alpha \), \( \alpha - x \) is positive and \( \beta - x \) is negative, making \( \frac{dx}{dt} < 0 \). Thus, the solutions move towards \( x = \alpha \), indicating stability for this interval.

Step 4: Finish Stability Analysis for the Last Interval

For \( x > \alpha \), both \( \alpha-x \) and \( \beta-x \) are negative, so \( \frac{dx}{dt} > 0 \). Therefore, solutions move towards \( x = \alpha \), confirming that \( x = \alpha \) is asymptotically stable, and \( x = \beta \) is unstable.

Key Concepts

Critical Points in Autonomous Differential Equations

In autonomous differential equations, critical points are values where the derivative (rate of change) of the system equals zero. This means the system stops changing at these points. Let’s look at the differential equation from the example: \[ \frac{d x}{d t} = k(\alpha-x)(\beta-x) \]Critical points occur where the equation becomes zero. Here, you'll set up the equation like this: \[ k(\alpha-x)(\beta-x) = 0 \]The solutions to this equation—where each part of the product is zero—give the critical points:

• \( x = \alpha \)
• \( x = \beta \)

Finding these points is fundamental as they help determine where the behavior of the system changes. Critical points signal potential equilibrium states where the rate of change pauses. Understanding these helps in analyzing how a solution behaves nearby.

Understanding Stability Analysis

Stability analysis is about finding out what happens near critical points. once we locate these points, the next step is to examine whether the solutions near these points tend to move away or towards them over time. In essence, we want to know if these points are places where the system stabilizes or not. Let's consider our critical points:

• For \(x = \beta\), the differential changes sign such that \( \frac{dx}{dt} > 0 \) when \(x < \beta\), implying that solutions move away. This makes \(x = \beta\) unstable.
• For \(x = \alpha\), \( \frac{dx}{dt} < 0 \) when \( \beta < x < \alpha \), switching to \(\frac{dx}{dt} > 0\) for \(x > \alpha\), indicating motion towards \(x = \alpha\). Therefore, \(x = \alpha\) is asymptotically stable.

Asymptotic stability means that when the system is near \(x = \alpha\), it tends to move closer until it stabilizes right there. Conversely, instability at \(x = \beta\) suggests that any small disturbance will cause a drift away from this point. Analyzing stability helps predict long-term behavior.

Introduction to First-order Differential Equations

First-order differential equations, like the one in our example, involve derivatives with respect to one variable and are fundamental in describing dynamic changes. The key feature of first-order equations is that they involve only the first derivative of the unknown function—thus being "first order."In our setup, the differential equation looks like:
\[\frac{d x}{d t} = k(\alpha-x)(\beta-x)\]Here, \(\frac{d x}{d t}\) represents how \(x\) changes over time \(t\). The function on the right side defines how these changes happen based on the current state (value of \(x\)).

• "Autonomous" refers to the fact that the equation doesn’t explicitly depend on time \(t\). The changes in \(x\) rely solely on its current value.
• The constants \(\alpha\), \(\beta\), and \(k\) dictate the system's behavior, controlling how quickly and in which direction the changes happen.

By understanding the structure of first-order differential equations, you can gain insights into how complex systems evolve over time, which is vital for modeling various natural and technological processes.