Problem 13 - Advanced Engineering Mathematics

Question

In Problems, find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=x^{2} e^{y} \\ &y^{\prime}=y\left(e^{x}-1\right) \end{aligned} $$

Step-by-Step Solution

Answer

The critical point is \((0, 0)\).

Step 1: Understanding Critical Points

Critical points of a system \[\begin{aligned}x^{\prime} &= f(x, y) y^{\prime} &= g(x, y)\end{aligned}\]occur when both \(x' = 0\) and \(y' = 0\). This implies solving the system of equations where both derivatives are set to zero.

Step 2: Solving for \(x' = 0\)

Set \(x' = x^2 e^y = 0\). For this to be zero, either \(x^2 = 0\) or \(e^y = 0\). Since \(e^y > 0\) for all \(y\), it must be that \(x^2 = 0\), implying \(x = 0\).

Step 3: Solving for \(y' = 0\)

Given \(y' = y(e^x - 1) = 0\), this is zero when either \(y = 0\) or \(e^x - 1 = 0\), which resolves to \(e^x = 1\). This means \(x = 0\).

Step 4: Finding All Critical Points

From the previous steps,1. \(x = 0\) from solving \(x' = 0\).2. \(x = 0\) and any \(y = 0\) from \(y' = 0\).Therefore, the critical point is at \((0, 0)\).

Key Concepts

Understanding Plane Autonomous Systems

Plane autonomous systems are a type of dynamical system represented in a two-dimensional space. They consist of a system of equations that govern how points in this plane evolve over time, without an explicit dependency on time itself.
Such systems are utilized to model various natural and engineered processes that change with respect to each other rather than against time.
Common examples include predator-prey models in ecology and competing species models.

• A plane autonomous system is typically expressed as two coupled differential equations.
• These equations relate the derivatives of two variables, usually denoted as \(x'\) and \(y'\).
• Being autonomous means the system is time-independent, which simplifies the analysis.

In the context of our problem, the system is given by:\[\begin{aligned}&x^{\prime}=x^{2} e^{y} \&y^{\prime}=y\left(e^{x}-1\right)\end{aligned}\]This represents the rate of change of \(x\) and \(y\) in the plane.

Differential Equations in Autonomous Systems

Differential equations are mathematical tools used to describe the rates of change of variables. In plane autonomous systems, they help us understand how variables interact over time to form trajectories and patterns in the phase plane.

• Each equation in the system corresponds to a derivative, indicating how one variable changes as a function of the others.
• By setting the derivatives to zero, we identify points where the system is at equilibrium—these are called critical points.

For our system:\[x^{\prime}=x^{2} e^{y}
y^{\prime}=y\left(e^{x}-1\right)\]The critical points occur when both \(x'\) and \(y'\) are zero simultaneously.

• Solving \(x' = 0\) gives \(x = 0\).
• Solving \(y' = 0\) gives two possibilities: \(y = 0\) or \(x = 0\).

These solutions lead us to identify the critical point as \((0, 0)\).

Stability Analysis of Critical Points

Stability analysis in the context of differential equations involves determining whether small perturbations or changes around the critical points will decay, persist, or grow with time. This helps us understand the behavior of the system near its equilibrium points.

• A stable point is one where perturbations decay over time, suggesting the system will return to equilibrium.
• An unstable point implies that perturbations grow, causing the system to move away from the equilibrium.
• Semi-stable or neutral points result in behavior that depends on the direction of perturbation.

To perform stability analysis, we usually compute the Jacobian matrix of the system at the critical points and analyze its eigenvalues.In our exercise, the critical point \((0, 0)\) can be analyzed by setting up such a matrix and examining the eigenvalues:\[J = \begin{pmatrix} \frac{\partial x'}{\partial x} & \frac{\partial x'}{\partial y} \ \frac{\partial y'}{\partial x} & \frac{\partial y'}{\partial y} \end{pmatrix}\]Solving this provides insights into whether \((0, 0)\) is stable, unstable, or neutral, helping us predict the behavior of trajectories near the critical point.