Question

Find the equilibrium solution of the following equations, make a sketch of the direction field, for \(t \geq 0,\) and determine whether the equilibrium solution is stable. The direction field needs to indicate only whether solutions are increasing or decreasing on either side of the equilibrium solution. $$y^{\prime}(t)=-\frac{y}{3}-1$$

Short answer

Answer: The equilibrium solution \(y = -3\) exhibits stable stability.

Step-by-step solution

Find the equilibrium solution

To find the equilibrium solution, set \(y'(t)\) equal to zero and solve for y: $$0=-\frac{y}{3}-1$$ Now, we solve for y: $$y=-3$$ So, the equilibrium solution is \(y=-3\).

Analyze the behavior of the solutions around the equilibrium solution

Now we will analyze the behavior of the solutions around the equilibrium solution \(y=-3\). To do this, take note of the sign of \(y'(t)\) for y values above and below the equilibrium solution. Take a value just below the equilibrium, e.g., \(y= -3.1\), and check the sign of \(y'(t)\): $$y'(t)=-\frac{-3.1}{3}-1>0$$ So, solutions below the equilibrium solution \(y=-3\) are increasing. Take a value just above the equilibrium, e.g., \(y= -2.9\), and check the sign of \(y'(t)\): $$y'(t)=-\frac{-2.9}{3}-1<0$$ So, solutions above the equilibrium solution are decreasing.

Sketch the direction field

Based on the analysis in step 2, we can draw the direction field. For \(t \geq 0\), we can draw horizontal arrows indicating whether the solutions are increasing (arrows pointing to the right) or decreasing (arrows pointing to the left) on either side of the equilibrium solution: 1. For \(y<-3\), the solutions are increasing, so we draw arrows to the right in that region. 2. At \(y=-3\), the solutions are constant as it's the equilibrium solution (flat line). 3. For \(y>-3\), the solutions are decreasing, so we draw arrows to the left in that region.

Determine the stability of the equilibrium solution

We can now determine the stability of the equilibrium solution based on the direction field. If the solutions "move away" from the equilibrium solution as time goes on, then the equilibrium solution is unstable. If solutions "move towards" the equilibrium solution as time goes on, then the equilibrium solution is stable. In this case, the solutions are increasing (i.e., moving towards) the equilibrium solution on the side \(y<-3\), and the solutions are decreasing (i.e., moving towards) the equilibrium solution on the side \(y>-3\). Therefore, the equilibrium solution \(y = -3\) is stable.

Key concepts

Differential Equations

Differential equations form the cornerstone of many scientific and engineering disciplines. They are equations that relate a function to its derivatives and are employed to model phenomena where the rate of change of a quantity is significant. Such equations express the relationship between varying quantities and their rates of change over time or space.

For instance, the equation from the exercise, \(y^{\prime}(t)=-\frac{y}{3}-1\), is a first-order linear differential equation, which means it involves the first derivative of \(y\) with respect to \(t\) and is proportional to \(y\) plus some function of \(t\). The solution to this type of equation provides us with a function \(y(t)\) describing the behavior or state of a system over time.

In the context of our example, the process of finding the equilibrium solution begins with setting the derivative equal to zero, to determine the constant solutions (if any) over the domain of interest.

Direction Field Sketching

Direction field sketching is a qualitative tool for visualizing the solutions of differential equations without solving them explicitly. It involves drawing arrows which represent the slope of the solution curve at any given point in the domain. These arrows, known collectively as the 'direction field', indicate how the system behaves as time progresses.

The approach taken in the exercise allows us to understand the trends or directions that the solution of the differential equation might take. By analyzing the behavior around the equilibrium solution, we can infer general characteristics of the solutions. For example, when \(y'(t) > 0\), the solution will be increasing, and conversely, when \(y'(t) < 0\), the solution will be decreasing. Sketching the direction field helps to visualize this: to the left of \(y=-3\), the arrows point right, showing increase, and to the right, the arrows point left, indicating decrease.

Equilibrium Solution Analysis

Analyzing the stability of equilibrium solutions is essential to understanding the long-term behavior of systems described by differential equations. Equilibrium solutions are those where the system, once there, remains there without any further change unless perturbed.

Stability comes into play when considering small deviations around the equilibrium: a 'stable' equilibrium will 'attract' nearby solutions, meaning the system will return to equilibrium after a disturbance. In contrast, an 'unstable' equilibrium will 'repel' solutions, causing them to move further away after any perturbation.

In the exercise, we found that the equilibrium solution \(y = -3\) is stable because solutions on either side are moving towards it over time. This conclusion arises from examining the signs of \(y'(t)\) just above and below the equilibrium value. By illustrating the tendency of solutions to move towards the equilibrium point from both directions, we can effectively demonstrate the concept of stability to students, helping them grasp this phenomena in a more tangible way.