# Equilibrium Solutions - Understand & Graphically Visualize

• Bashyboy
In summary, equilibrium solutions are when the derivative of y with respect to x is equal to 0. Graphically, this is represented by little horizontal lines along the line y=y0. However, this does not necessarily mean that the function has a maximum or minimum at the equilibrium solution. To determine the stability of the equilibrium solution, you must plot points above and below the solution and check the direction of the solutions. Typically, equilibrium solutions do not correspond to actual extrema of non-equilibrium solutions, and solutions will asymptotically approach the equilibrium as t goes to infinity or minus infinity.
Bashyboy
Hello,

I want to make certain of my understanding of equilibrium solutions. Are equilibrium solutions the value(s) of y such that $\frac{dy}{dt} = 0$? So, suppose y = y0 is one of those solutions. Graphically, does this mean that one the horizontal line y = y0, little slope lines are too horizontal; and does this correspond to where the solution curves attain a maximum or a minimum?

The equilibrium solutions are when dy/dx=0. Graphically it means the all along the line y=y0, there are little horizontal lines (slope of 0). Just because the equilibrium solution is y=y0 does not mean that the function obtains a minimum or a maximum. You must plot points below and above the equilibrium solution to check the direction of where the solutions are going, hence the term to describe the graph as a direction field. Let's say you set y=1+y0(assuming y>0) and plug it into the differential equation(dy/dx) and the number you get is positive. Then you plug in y=y0-1 into the differential equation and get a positive munber. This would mean the there would not be a max or a min obtained. And it would mean that the equilibrium solution y=y0 is semistable. If you were to get opposite slope values above and below the equilibrium solution, then you would obtain either a max or a min.

Typically equilibrium solutions will not correspond to actual extrema of non-equilibrium solutions. If your differential equation is reasonably well behaved then given a point there is a unique solution passing through it - if that solution is the equilibrium solution then it means no other solution is passing through it. Typically solutions will asymptotically approach the equilibrium either as t goes to infinity or minus infinity (or both, depending on the differential equation)

## 1. What is an equilibrium solution?

An equilibrium solution is a point or set of points in a system where the behavior of the system remains constant over time. This means that the system has reached a balance between the forces or factors that are acting on it.

## 2. How do you graphically visualize an equilibrium solution?

To graphically visualize an equilibrium solution, you can plot the equations or functions that represent the system and look for points where the system remains constant or where the slope of the graph is zero. These points are the equilibrium solutions.

## 3. What are the types of equilibrium solutions?

There are two types of equilibrium solutions: stable and unstable. A stable equilibrium solution is one where the system returns to the solution after being slightly disturbed, while an unstable equilibrium solution is one where the system moves away from the solution after being slightly disturbed.

## 4. How do you determine the stability of an equilibrium solution?

The stability of an equilibrium solution can be determined by analyzing the behavior of the system around the solution. If the system returns to the solution after being slightly disturbed, it is a stable equilibrium solution. If the system moves away from the solution after being slightly disturbed, it is an unstable equilibrium solution.

## 5. Why is understanding equilibrium solutions important in science?

Understanding equilibrium solutions is important in science because many real-world systems, such as chemical reactions, ecosystems, and economic markets, can be modeled and analyzed using the concept of equilibrium. By understanding the behavior of these systems at equilibrium, we can make predictions and manipulate the systems to achieve desired outcomes.

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