Question

What can you say about the behavior of the solution? Are there equilibria? What stability do these equilibria have? $$y^{\prime }=y-x$$

Short answer

The solution exhibits stable behavior along the line $y=x$; this line is the stable equilibrium.

Step-by-step solution

Identify Equilibrium Points

To find the equilibria, we set the derivative equal to zero: $y^{\prime }=y-x=0$. Thus, the equilibria occur when $y=x$.

Analyze Stability of Equilibria

We use the first derivative test to determine stability. The objective is to determine the sign of $y^{\prime }$ around equilibrium points. - If $y<x$, then $y^{\prime }=y-x<0$, indicating that $y$ decreases.- If $y>x$, then $y^{\prime }=y-x>0$, indicating that $y$ increases.Therefore, in both cases, the solutions move towards $y=x$. This suggests that the equilibrium points are stable.

Conclude on Solution Behavior

The solutions of the system are attracted to the line $y=x$. At any point off the line, the system will adjust such that $y$ moves towards $x$, indicating that the equilibrium line $y=x$ is stable.

Key concepts

Equilibrium Points

Equilibrium points are the foundation of understanding the behavior of a dynamical system. These are points where the system is at rest, meaning there is no change over time. This happens when the derivative of the system's equation is equal to zero. In our exercise, we are dealing with the differential equation $y^{\prime }=y-x$.The equilibrium occurs where $y^{\prime }=0$. By solving $y=x$, we find the equilibrium points where the system is perfectly balanced. Equilibrium points are crucial because they tell us about the constant solutions of the equation where changes are not happening. These points help us predict the long-term behavior of the system, as they are potential rests or steady states. Finding equilibrium points is like solving a puzzle to see where pieces fit without any further movement. Once found, these points serve as a reference for analyzing how the system might behave overall.

Stability Analysis

After finding equilibrium points, the next step is to analyze their stability. Stability analysis tells us how solutions behave when they are close to the equilibrium points. If a small change or disturbance occurs, will the system return to equilibrium or move away? In the exercise, stability is analyzed by looking at the sign of the derivative $y^{\prime }$ around the equilibrium points. - If $y<x$, then $y^{\prime }=y-x<0$, indicating that $y$ decreases.- If $y>x$, then $y^{\prime }=y-x>0$, which means $y$ increases.In both scenarios, $y$ moves towards $x$, which indicates that equilibrium is stable. The concept of stability is important because it helps predict whether any slight change in conditions will affect the overall system significantly or not. Stability analysis reveals if an equilibrium point is like a valley (stable) or peak (unstable), guiding how solutions evolve near these points.

Dynamical Systems

Dynamical systems are used to model the changing state of systems over time. They can be seen in various fields, from physics to biology, as they describe how one or more quantities evolve. The differential equation $y^{\prime }=y-x$ represents a simple dynamical system describing how $y$ changes relative to $x$.By understanding dynamical systems, we can capture the essence of how components interact and evolve given certain rules or conditions, such as the differential equation in our case. This equation and its solution show how the state of the system moves towards its equilibrium, specifically along the line $y=x$. The study of dynamical systems often involves:

• Identifying equilibrium points to find where no change occurs.
• Performing stability analysis to understand what happens near those points.
• Exploring solution behaviors to see how the system evolves over time.

Understanding dynamical systems allows scientists and engineers to predict future behavior, model phenomena, and control processes in everyday life.