Problem 4 - Advanced Engineering Mathematics

Question

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. $$\frac{d x}{d t}=-k x \ln \frac{x}{k}, x>0$$

Step-by-Step Solution

Answer

The critical point \(x = k\) is asymptotically stable.

Step 1: Identify Critical Points

For the autonomous differential equation \( \frac{d x}{d t} = -k x \ln \frac{x}{k} \), the critical points are found by setting the right-hand side equal to zero. This gives us \(-k x \ln \frac{x}{k} = 0\). Since \(x > 0\) and \(k > 0\), this simplifies to \(\ln \frac{x}{k} = 0\). Solving \(\ln \frac{x}{k} = 0\) gives \(x = k\). Thus, the critical point is \(x = k\).

Step 2: Determine Stability Using Derivative

To determine the stability of the critical point, we consider the derivative of the right-hand side of the differential equation with respect to \(x\). Calculate \(f'(x)\) where \(f(x) = -k x \ln \frac{x}{k}\). Derivative gives \(f'(x) = -k (1 + \ln \frac{x}{k})\). Analyze \(f'(x)\) at the critical point \(x = k\), giving \(-k (1 + \ln 1) = -k(1+0) = -k\). Since \(-k < 0\), the critical point \(x = k\) is asymptotically stable.

Key Concepts

Critical Points

Critical points in the context of autonomous differential equations are those values of the variable that make the derivative zero. These are essentially the points where the system is in equilibrium. For the given differential equation \( \frac{d x}{d t} = -k x \ln \frac{x}{k} \), identifying critical points is one of the first steps in understanding the behavior of the system. To find critical points, we set the right-hand side of the equation to zero:

• \(-k x \ln \frac{x}{k} = 0\)

Given that \(x > 0\) and \(k > 0\), the equation simplifies to \(\ln \frac{x}{k} = 0\), and solving this gives \(x = k\). This means the critical point for the system is \(x = k\), where the system doesn't change over time, representing a kind of steady state or balance.

Stability Analysis

Stability analysis is the process of determining whether small deviations from a critical point will return to that point, or whether they will grow with time. In simpler terms, it's about understanding if a system is capably 'holding steady' when nudged a bit. Once the critical points are identified, we evaluate if they are stable, unstable, or asymptotically stable by examining the sign of the derivative at those points. This involves:

• Finding the derivative of the right-hand side function of the differential equation.
• Substituting the critical point into this derivative.
• Checking the sign of the result.

If the derivative \(f'(x)\) evaluated at the critical point is negative, the point is stable. If positive, it is unstable. For the equation given, \(f'(x) = -k (1 + \ln \frac{x}{k})\). Evaluating this at \(x = k\) shows that \(f'(k) = -k\), which is negative, indicating that the critical point is stable.

Asymptotic Stability

Asymptotic stability is a specific type of stability pertaining to differential equations. It indicates that not only does the system return to a critical point after a small disturbance, but it does so by shrinking towards it asymptotically over time. In mathematical terms, if \(x(t)\) is a solution to the differential equation and \(x = k\) is a critical point, then \(x(t)\) approaches \(k\) as \(t\) goes to infinity. In our exercise:

• \(f'(x) = -k(1+ \ln \frac{x}{k})\) evaluated at the critical point \(x = k\) yields \(-k\), which is less than zero.
• Since \(-k < 0\), disturbances from \(x = k\) will cause the state to return to \(k\).

Thus, we conclude that the critical point \(x = k\) is not only stable but also asymptotically stable.

Derivative

Derivatives are fundamental tools in calculus that describe the rate at which one quantity changes with respect to another. In the context of differential equations, they often describe how the state of a system evolves over time.For the autonomous differential equation given, the derivative of the function \(f(x) = -k x \ln \frac{x}{k}\) with respect to \(x\) is a vital part of analyzing stability:

• \(f(x) = -k x \ln \frac{x}{k}\)
• The derivative, \(f'(x) = -k (1 + \ln \frac{x}{k})\), tells us how the function \(f(x)\) changes as \(x\) changes.

This derivative helps us determine the stability of the critical point \(x = k\) by looking at the sign of \(f'(k)\). Since \(f'(x)\) gives insights into the behavior near the critical point, it is a crucial part of stability analysis.

First-Order Differential Equations

First-order differential equations involve derivatives of the first degree, usually describing systems where the change in a variable depends only on the current state and not on any higher derivatives. Such equations form the mathematical backbone for modeling a wide range of phenomena.In the provided exercise, the differential equation \(\frac{d x}{d t} = -k x \ln \frac{x}{k}\) is an example of a first-order autonomous differential equation, where:

• 'First-order' refers to the presence of only the first derivative, \(\frac{d x}{d t}\).
• 'Autonomous' suggests that the equation doesn’t explicitly depend on the independent variable \(t\).

Analyzing these equations allows us to predict how systems evolve, making them fundamental in understanding behaviors across physics, biology, and economics.