Problem 13 - Advanced Engineering Mathematics

Question

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 only critical point is (0, 0).

Step 1: Setting the System to Zero

To find the critical points of the system, we need to set both differential equations to zero. This means solving the system of equations:\[ x^2 e^y = 0 \]\[ y(e^x - 1) = 0 \]

Step 2: Solving the First Equation

The first equation \( x^2 e^y = 0 \) implies either \( x^2 = 0 \) or \( e^y = 0 \). Since the exponential function \( e^y \) is never zero, we must have \( x^2 = 0 \). Thus, \( x = 0 \).

Step 3: Solving the Second Equation

The second equation is \( y(e^x - 1) = 0 \). This gives us two cases: either \( y = 0 \) or \( e^x - 1 = 0 \).- **Case 1**: If \( y = 0 \), the equation is satisfied regardless of the value of \( x \).- **Case 2**: If \( e^x - 1 = 0 \), then \( e^x = 1 \), which implies \( x = 0 \), because \( e^0 = 1 \).

Step 4: Finding Critical Points

From the above findings:- Combining \( x = 0 \) from Step 2 with \( y = 0 \) from Case 1 in Step 3, we have the critical point (0, 0).- From Case 2 in Step 3, using \( x = 0 \), we confirm that the condition \( y \) can be any value, but since this results in only reinforcing (0, 0), no additional critical points arise.

Key Concepts

Autonomous Systems

In the world of mathematics, an autonomous system is a type of differential equation. It's special because the rules guiding its behavior do not change with time. This means the system's equations depend solely on the variables involved, like "x" and "y" in our given plane system, but not on time or another independent variable.

For instance, in our provided exercise, we have the system:

• \( x' = x^2 e^y \)
• \( y' = y(e^x -1) \)

These don't change over time, but rather depend on the values of \( x \) and \( y \) themselves.

Autonomous systems often model real-world situations where the current state determines what happens next, independent of any external timing or sequence. They are crucial in many fields, such as biology for modeling population dynamics, physics for understanding mechanical systems, and economics for simulating market equilibria.

Differential Equations

Differential equations are equations that relate a function with its derivatives. They describe how a certain quantity changes over time, or in our case, how it changes with respect to other variables. In simpler terms, they tell us how the rate of change of a variable depends on the variable itself and potentially other variables.

In the given exercise, we have two differential equations:

• The first one, \( x' = x^2 e^y \), shows how the rate of change of \( x \) is dependent on both \( x \) and \( y \). If \( x \) were zero, the rate of change would stop because of multiplication by zero.
• The second one, \( y' = y(e^x - 1) \), similarly shows that the rate of change of \( y \) depends on the value of \( y \) and \( x \).

Solving differential equations is crucial for finding critical points where these rates are zero. This means the system is in a state of equilibrium and no change occurs. In this exercise, solving the equations help us locate these critical points at \( (0, 0) \).

Plane Systems

In mathematics, plane systems refer to differential systems composed of two equations that define how two variables, \( x \) and \( y \), interact with each other. They are often studied in a two-dimensional plane or coordinate system. The plane allows us to visually represent the interaction and changes between variables.

In our exercise, the system is explored in terms of a plane system:

• Each axis corresponds to one of the variables \( x \) or \( y \).
• The equations determine how these variables move in relation to each other in this plane.

The critical points in plane systems are particularly important because they represent locations in the plane where the system's behavior changes or stops (equilibriums). When both equations of the system equal zero, we find these critical points. In this exercise's system, the point \( (0, 0) \) is such a point, indicating that when both \( x \) and \( y \) are zero, the system reaches a state of no change or equilibrium.