Problem 5 - 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. $$ \frac{d T}{d t}=k\left(T-T_{0}\right) $$

Step-by-Step Solution

Answer

The critical point \( T = T_0 \) is asymptotically stable.

Step 1: Identify the Critical Points

To determine the critical points of the differential equation \( \frac{d T}{d t} = k(T-T_0) \), we need to set the derivative equal to zero: \( k(T-T_0) = 0 \). Solving this, we find that the critical point is \( T = T_0 \).

Step 2: Analyze Stability Using the Sign of the Derivative

To classify the stability of the critical point, we consider the sign of the derivative \( \frac{dT}{dt} \) around \( T = T_0 \). The equation simplifies to \( k(T-T_0) \), where \( k \) is positive by assumption. For \( T > T_0 \), \( T-T_0 > 0 \), so \( \frac{dT}{dt} > 0 \) indicating \( T \) is increasing. For \( T < T_0 \), \( T-T_0 < 0 \), so \( \frac{dT}{dt} < 0 \) indicating \( T \) is decreasing.

Step 3: Classify the Critical Point

Since \( \frac{dT}{dt} > 0 \) for \( T > T_0 \) and \( \frac{dT}{dt} < 0 \) for \( T < T_0 \), any deviation from \( T_0 \) will result in the system moving back towards \( T_0 \). This means that the critical point \( T = T_0 \) is asymptotically stable.

Key Concepts

Critical Points

In first-order autonomous differential equations, a critical point is a value where the derivative of the dependent variable is zero. Specifically, in the equation \( \frac{dT}{dt} = k(T - T_0) \), setting \( \frac{dT}{dt} = 0 \) allows us to solve for \( T \). This is because critical points occur where no change is happening over time, indicating equilibrium. By solving \( k(T - T_0) = 0 \), we find that \( T = T_0 \) is the critical point. The importance of identifying critical points lies in their ability to give insight into the system's long-term behavior, as they often represent stable or unstable equilibrium positions.

Stability Analysis

Stability analysis helps us understand whether a critical point will attract or repel trajectories close to it. For the differential equation \( \frac{dT}{dt} = k(T - T_0) \), we assess the system by examining the sign of \( \frac{dT}{dt} \) around the critical point \( T = T_0 \).

• For \( T > T_0 \), the term \( T - T_0 \) is positive, making \( \frac{dT}{dt} > 0 \). This suggests that \( T \) increases, moving away from equilibrium if perturbed upwards.
• For \( T < T_0 \), \( T - T_0 \) is negative, so \( \frac{dT}{dt} < 0 \), indicating that \( T \) decreases, returning towards the equilibrium if perturbed downwards.

By this analysis, since in both cases \( T \) moves back toward \( T_0 \), the critical point \( T = T_0 \) is asymptotically stable. Asymptotically stable points tend to restore equilibrium over time, even if the system experiences small disturbances.

First-Order Differential Equations

First-order differential equations involve derivatives of the first degree and are often used to describe various natural phenomena such as population dynamics, thermal changes, or electrical circuits. An autonomous differential equation is a specific type of first-order equation where the rate of change of the dependent variable is given by a function of that variable alone, like in our equation \( \frac{dT}{dt} = k(T - T_0) \). In such equations, time does not explicitly appear in the function.
Understanding these models is crucial as they not only describe systems changing over time but also help predict future behavior. Initial conditions, such as the value of \( T \) at a particular point in time, play an essential role in solving these equations, providing unique solutions that chart the trajectory over time from that initial starting point.