Problem 6 - 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. $$ m \frac{d v}{d t}=m g-k v $$

Step-by-Step Solution

Answer

The critical point \( v = \frac{mg}{k} \) is asymptotically stable.

Step 1: Identify the Differential Equation

The given differential equation is \( m \frac{dv}{dt} = mg - kv \). This is a first-order autonomous differential equation.

Step 2: Set the Derivative to Zero

To find the critical points, we set the derivative equal to zero: \( m \frac{dv}{dt} = 0 \). This simplifies to \( mg - kv = 0 \).

Step 3: Solve for the Critical Point

Solving \( mg - kv = 0 \) for \( v \), we get \( v = \frac{mg}{k} \). This is the critical point of the equation.

Step 4: Analyze the Stability of the Critical Point

To determine the stability of the critical point, examine the sign of the derivative of the right-hand side with respect to \( v \): \( \frac{d}{dv}(mg - kv) = -k \). Since \(-k < 0\) (as \(k\) is positive), the critical point is asymptotically stable.

Key Concepts

Critical Points

When dealing with first-order autonomous differential equations, identifying critical points is essential. Critical points are values where the rate of change, or the derivative, is zero. This happens because at these points, the system is in a state of equilibrium.

To find the critical points in the equation provided, we set the derivative equal to zero: \[m \frac{dv}{dt} = mg - kv = 0\].
By solving this equation, we find the critical point at:
\[v = \frac{mg}{k}\].
This value of \(v\) is where the system neither increases nor decreases, representing a balance between the forces described by the equation.

• Critical points help predict the behavior of a system over time.
• If an initial condition is set at a critical point, theoretically, the system should remain constant over time.

Understanding critical points is crucial in many fields such as physics, engineering, and biology, where systems often reach equilibrium states.

Asymptotic Stability

Asymptotic stability refers to the behavior of a system's solutions as time approaches infinity. A critical point is considered asymptotically stable if solutions of the differential equation tend to reach the critical point over time.

In the context of the differential equation \(m \frac{dv}{dt} = mg - kv\), once we identified the critical point \(v = \frac{mg}{k}\), we can determine its stability. The critical point's stability is analyzed by examining the derivative of the equation with respect to \(v\):
\[\frac{d}{dv}(mg - kv) = -k\].
Since \(-k < 0\) and \(k\) is assumed positive, this indicates that the derivative is negative, demonstrating that the critical point is asymptotically stable.

• When the critical point is asymptotically stable, small perturbations or deviations from the point will diminish over time, leading the system back to equilibrium.
• This property is highly desirable in real-world applications where maintaining system stability is crucial.

First-Order Differential Equations

First-order differential equations involve derivatives of only the first degree. They are often expressed in the form \(\frac{dy}{dx} = f(x, y)\), where the highest derivative is the first.

In our specific exercise, the equation is \(m \frac{dv}{dt} = mg - kv\), which is a first-order differential equation. These equations can be classified as autonomous or non-autonomous:

• Autonomous: The rate of change depends solely on the dependent variable, not the independent variable (time \(t\) in this case), as seen in the given equation.
• Non-autonomous: The derivative depends on both the independent and dependent variables.

First-order differential equations are crucial in modeling continuous dynamical processes. They provide insights into phenomena like growth rates, decay processes, and oscillatory systems.