Problem 21 - Advanced Engineering Mathematics

Question

In Problems 21-26, classify (if possible) each critical point of the given second-order differential equation as a stable node, a stable spiral point, an unstable spiral point, an unstable node, or a saddle point. $$ \theta^{\prime \prime}=(\cos \theta-0.5) \sin \theta, \quad|\theta|<\pi $$

Step-by-Step Solution

Answer

Critical points are at \(0 \ - \text{saddle point,} \ \pm \frac{\pi}{3} \ - \text{stable nodes,} \ \pi, -\pi \ - \text{saddle points.}\)

Step 1: Identify the Critical Points

To find the critical points, set \(\theta'' = 0\). This leads to the equation \((\cos \theta - 0.5) \sin \theta = 0\). We solve for \(\theta\), giving us two conditions: \(\cos \theta = 0.5\) or \(\sin \theta = 0\). Solving these, we get \(\theta = 0, \pm \pi\), and \(\theta = \pm \frac{\pi}{3}\) as critical points.

Step 2: Classify Each Critical Point

To classify, compute the second derivative of the function f with respect to \(\theta\). Here, \(f(\theta) = (\cos \theta - 0.5) \sin \theta\). Compute \(f'(\theta)\) to analyze stability. \(f'(\theta) = -\sin^2\theta -\cos \theta \sin \theta + 0.5 \sin \theta \). Evaluate \(f'(\theta)\) at the critical points.

Step 3: Evaluate Stability at Critical Points

\(\theta = 0\): \(f'(0) = -\sin^2(0) = 0\), indicates a saddle point since further testing for higher derivatives is needed but balance condition fails. \(\theta = \pm \frac{\pi}{3}\): \(f'(\pm \frac{\pi}{3}) \approx -0.5\), indicating stable nodes (since \(f' < 0\)). \(\theta = \pi, -\pi\): These points suggest continuity issues or regular moments in phase, lean effects as saddle due to setup method.

Key Concepts

Second-order Differential Equations

Second-order differential equations are equations that involve the second derivative of a function, typically representing a rate of change related to another rate of change. In the context of our exercise, the second-order differential equation given is \( \theta'' = (\cos \theta - 0.5) \sin \theta \). These types of equations are fundamental in describing physical systems, such as oscillations and dynamics, where acceleration or force needs to be modeled.

When tackling second-order differential equations, finding solutions typically involves identifying critical points by setting the second derivative to zero. This step helps to pinpoint equilibrium states of the system. For instance, in this problem, the equation was simplified to \( (\cos \theta - 0.5) \sin \theta = 0 \) to find those critical points, such as \( \theta = 0, \pm \pi \) and \( \theta = \pm \frac{\pi}{3} \).

Working with these equations requires understanding both the mathematical methodology and the physical implications, making them a central component of studies in physics and engineering.

Stability Classification

Stability classification refers to determining the nature of equilibrium points, particularly whether they are stable or unstable. Stability is crucial when assessing how a system behaves near its critical points. In this exercise, the stability type of each critical point is identified by examining the first derivative of the function associated with the differential equation.

To classify stability:

• If \( f'(\theta) < 0 \), the critical point is stable, indicating perturbations will return to equilibrium.
• If \( f'(\theta) > 0 \), the critical point is unstable, suggesting perturbations will move away from equilibrium.
• Saddle points occur if the first test is inconclusive, often requiring further derivative tests.

In our solution, for example, \( \theta = \pm \frac{\pi}{3} \) was determined to be a stable node as \( f'(\pm \frac{\pi}{3}) \approx -0.5 \). Saddle points emerged at \( \theta = 0 \) and \( \theta = \pm \pi \), hinting at more complex stability behaviors.

Phase Plane Analysis

Phase plane analysis is a graphical method to study second-order differential equations. It involves plotting trajectories of a dynamic system's states, offering a visual perspective of the system's behavior over time. A phase plane is typically plotted with the variable \( \theta \) along one axis and its derivative \( \theta' \) on the other axis.

This analysis helps visualize how different initial conditions in a system evolve. By observing the trajectories, we can identify patterns like spirals, nodes, and saddle points, corresponding to different stability types. In our problem, the phase plane helps classify the behavior around critical points like stable nodes at \( \theta = \pm \frac{\pi}{3} \) and saddle points at \( \theta = 0 \).

The phase plane is a powerful tool that simplifies the understanding of complex dynamic systems, aiding in the classification and prediction of system behaviors, crucial in fields like control systems and mechanical vibrations.