Problem 14 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

The critical points are \( (n\pi, n\pi) \) for integer \( n \).

Step 1: Define Critical Points

To find the critical points for the autonomous system, we need to set both derivatives equal to zero. Thus, we have: \( x' = \sin y = 0 \) and \( y' = e^{x-y} - 1 = 0 \).

Step 2: Solve for \( \sin y = 0 \)

The equation \( \sin y = 0 \) will be satisfied when \( y = n\pi \), where \( n \) is an integer number. This gives the points where the derivative of \( x \) is zero.

Step 3: Solve for \( e^{x-y} - 1 = 0 \)

For \( e^{x-y} - 1 = 0 \), solve the equation by rearranging it to: \( e^{x-y} = 1 \). This implies \( x-y = 0 \), hence \( x = y \).

Step 4: Combine the Solutions

Substitute \( y = n\pi \) from Step 2 into \( x = y \) obtained in Step 3. This gives \( x = n\pi \) as well. Therefore, the critical points are \((n\pi, n\pi)\), where \( n \) is an integer.

Key Concepts

Critical Points

Critical points in a plane autonomous system are locations where the system's derivatives are zero. These points represent a state where the system does not change, making them important for understanding the behavior of the system. To find critical points, set the equations for the derivatives equal to zero. For example, in the given system, we find:

• \( x' = \sin y = 0 \)
• \( y' = e^{x-y} - 1 = 0 \)

Finding the values of \( x \) and \( y \) that satisfy these equations helps identify the critical points of the system. In this case, solving these equations gives us critical points of the form \((n\pi, n\pi)\), where \( n \) is an integer. This method is a staple in the analysis of differential equations.

Plane System

A plane system refers to a set of equations that describe how two variables change over time. The variables, typically \(x\) and \(y\), form an ordered pair that can be plotted on a two-dimensional plane. This allows us to visualize the system's dynamics.
For the given system:

• The first equation \( x' = \sin y \) dictates how \( x \) changes based on \( y \).
• The second equation \( y' = e^{x-y} - 1 \) shows how \( y \) changes based on both \( x \) and \( y \).

By analyzing these coupled equations, one can gain insights into how different initial conditions and parameter values affect the system’s evolution over time. Ultimately, the graphical representation in the plane allows for a more intuitive understanding of the complex relationships between \( x \) and \( y \).

Differential Equations

Differential equations are equations that involve the derivatives of a function. They play a pivotal role in modeling a variety of real-world phenomena, describing how a system changes over time.In the exercise, the differential equations are:

• \( x' = \sin y \)
• \( y' = e^{x-y} - 1 \)

These equations express the rate of change of \( x \) and \( y \) as components of another system. To solve a differential equation means to find, for any given point, the direction in which the system moves. These solutions can be expressed as functions depicting the relation between the variables and time or as critical points where the system remains stationary.

Phase Plane Analysis

Phase plane analysis is a technique used to study autonomous systems of two differential equations. It involves plotting the variables in a phase plane to visually analyze the system's behavior across time.In a phase plane, critical points can be identified as intersections where trajectories do not change, such as the points \((n\pi, n\pi)\) found earlier.

• These points help describe the system's stability and identify equilibrium states.
• They allow for the classification of system behavior around these points, whether spiraling, oscillatory, or stabilizing.
• Graphical representations in phase planes provide an intuitive view of the long-term behavior of solutions.

Through this type of analysis, one can better understand the global dynamics of the system and predict its future states, which is immensely useful in fields such as physics, biology, and economics.