Problem 10 - 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. $$\frac{d A}{d t}=k \sqrt{A}(K-\sqrt{A}), A>0$$

Step-by-Step Solution

Answer

The critical point \( A = K^2 \) is asymptotically stable.

Step 1: Identify Critical Points

To find the critical points of the differential equation \( \frac{dA}{dt} = k \sqrt{A}(K - \sqrt{A}) \), set the right side to zero: \[ k \sqrt{A}(K - \sqrt{A}) = 0. \] This gives two conditions: 1) \( \sqrt{A} = 0 \), which means \( A = 0 \), but \( A > 0 \) by assumption. Hence, \( A = 0 \) is not considered.2) \( K - \sqrt{A} = 0 \), which means \( \sqrt{A} = K \) or \( A = K^2 \).Thus, the critical point is \( A = K^2 \).

Step 2: Determine Stability Using Derivative Test

The stability of the critical point can be determined by examining the derivative of the function. Differentiating \( f(A) = k \sqrt{A}(K - \sqrt{A}) \) with respect to \( A \), we get:\[ f'(A) = \frac{k}{2\sqrt{A}}(K) - \frac{k}{2}(1) = \frac{kK}{2\sqrt{A}} - \frac{k}{2}. \]Evaluate \( f'(A) \) at the critical point \( A = K^2 \):\[ f'(K^2) = \frac{kK}{2K} - \frac{k}{2} = \frac{k}{2} - \frac{k}{2} = 0. \]Since the derivative test yields \( f'(K^2) = 0 \), check the sign of \( f'(A) \) around \( A = K^2 \) to classify stability.

Step 3: Check Local Behavior for Stability

To determine stability, examine the sign of \( f'(A) \) around the critical point. For \( A < K^2 \) and \( A > K^2 \):- For \( A < K^2 \), \( \sqrt{A} < K \), so both terms \( k \sqrt{A} \) and \( K - \sqrt{A} \) are positive, meaning \( \frac{dA}{dt} > 0 \).- For \( A > K^2 \), \( \sqrt{A} > K \), so \( \frac{dA}{dt} < 0 \). With \( \frac{dA}{dt} > 0 \) to the left and \( < 0 \) to the right of \( A = K^2 \), this indicates an asymptotically stable point.

Key Concepts

Stability Analysis

In differential equations, stability analysis helps us determine whether the solutions of an equation remain close to a critical point, known as an equilibrium point, over time. Specifically, we distinguish whether these points are stable, asymptotically stable, or unstable.
For a critical point to be stable, small perturbations or changes in initial conditions should not cause the system to diverge significantly from that point.
Asymptotic stability goes a step further and involves solutions not only remaining near the critical point but also returning to it over time if they are slightly perturbed.
If a critical point is unstable, even small perturbations can cause solutions to drift far away from the said point.
The nature of the critical points can often be determined by examining how solutions to the differential equation behave as they approach or move away from the point of equilibrium. Understanding which factors contribute to the stability or instability of these points can assist in predicting long-term behaviors of systems modeled by these equations, making stability analysis an essential part of understanding differential equations.

First-Order Autonomous Differential Equations

First-order autonomous differential equations are equations wherein the rate of change of a variable is defined only by the state of the system itself, and not directly by time. For an equation like \( \frac{dA}{dt} = f(A) \), there is no explicit time-dependence.
Such equations are quite powerful in modeling real-world systems where the dynamics depend solely on the current state, such as population growth or chemical reactions.
The beauty of these equations lies in their simplicity; they describe how systems evolve solely based on their own characteristics.
This makes them easier to analyze compared to those with explicit time variation.A critical part of working with autonomous differential equations is understanding the concept of equilibrium or critical points, where the system does not change over time, that is, when \( f(A) = 0 \).
Finding these points is often the first step in analyzing such equations, as it indicates where the state of the system is neither increasing nor decreasing.Equilibrium analysis is closely tied to determining the stability of these systems, which tells us how the system will behave around these points.

Derivative Test for Stability

The derivative test for stability is a common method used to determine whether a critical point of a function is stable, asymptotically stable, or unstable.It involves evaluating the derivative \( f'(A) \) of the function \( f(A) \) at the critical point.
If \( f'(A) < 0 \) at the critical point, the point is usually asymptotically stable since the function is decreasing, attracting trajectories towards the point.
If \( f'(A) > 0 \), the point is considered unstable as any small perturbation will result in the state moving away from the critical point, indicating instability.When \( f'(A) = 0 \), as in our case with \( f'(K^2) = 0 \), the test is inconclusive. You need further analysisby checking the signs around the critical point to conclude.Analyzing the behavior on either side of the critical pointprovides the necessary insights,for instance checking if \( \frac{dA}{dt} \) is positive or negative as you shift away from the point. This approach ensures we thoroughly classify the point's stability by observing local dynamics rather than purely relying on derivative's direct analysis.