Problem 19 - Applied Calculus

Question

(a) What are the equilibrium solutions for the differential equation $$\frac{d y}{d t}=0.2(y-3)(y+2) ?$$ (b) Use a graphing calculator or computer to sketch a slope field for this differential equation. Use the slope field to determine whether each equilibrium solution is stable or unstable.

Step-by-Step Solution

Answer

Equilibrium solutions are \(y = 3\) (unstable) and \(y = -2\) (stable).

Step 1: Understanding Equilibrium Solutions

Equilibrium solutions occur when the derivative \(\frac{dy}{dt} = 0\). These are the values of \(y\) for which the rate of change is zero, meaning the system is at rest at these points.

Step 2: Find Equilibrium Solutions by Setting Derivative to Zero

Set the differential equation equal to zero and solve for \(y\): \[ 0.2(y-3)(y+2) = 0 \]. This equation is zero if either \(y-3 = 0\) or \(y+2 = 0\). Solve these equations to find the equilibrium solutions \(y = 3\) and \(y = -2\).

Step 3: Sketch Slope Field Using Graphing Tool

Using a graphing calculator or computer software, input the differential equation \(\frac{dy}{dt} = 0.2(y-3)(y+2)\). Generate the slope field, which displays small line segments with slopes corresponding to the derivative at various points in the plane.

Step 4: Analyze Stability from Slope Field

Examine the behavior of the slope field near each equilibrium solution. At \(y = 3\), observe the segments slope away, indicating this is an unstable equilibrium. At \(y = -2\), the slope segments point towards \(y = -2\), indicating this is a stable equilibrium.

Key Concepts

Differential Equations

Differential equations are mathematical expressions that describe how quantities change in relation to each other. They involve derivatives, which represent rates of change. In the given problem, the equation \( \frac{dy}{dt} = 0.2(y-3)(y+2) \) tells us how the variable \( y \) changes over time \( t \).
When working with differential equations, equilibrium solutions are of particular interest. These occur where the change stops, meaning the derivative equals zero. To find them, you set the right side of the equation to zero and solve for \( y \). In this case, \( 0.2(y-3)(y+2) = 0 \) gives the solutions \( y = 3 \) and \( y = -2 \).
These solutions reveal where the system is "at rest" or unchanging over time, which can help predict how the system behaves under various conditions.

Slope Field

A slope field, also called a direction field, is a graphical representation of a differential equation. It consists of small line segments or arrows in the plane that indicate the slope of the solution curve at that point. This visual tool helps you understand the behavior of solutions to differential equations without solving them explicitly.
To create a slope field for the equation \( \frac{dy}{dt} = 0.2(y-3)(y+2) \), you use software or a graphing calculator. The tool will plot a collection of lines that each reflect the slope given by the differential equation at various points. Such a display allows you to visualize how the system evolves over time.
Observing the slope field around the equilibrium points \( y = 3 \) and \( y = -2 \), you can infer the stability of these solutions based on the directions of the slopes.

Stability Analysis

Stability analysis is used to determine whether small deviations from the equilibrium solutions grow or diminish over time. For each equilibrium solution, you can analyze the surrounding slope field to determine if it is stable or unstable.
In our example, the equilibrium at \( y = 3 \) shows line segments sloping away from it. This suggests that if the system is slightly perturbed, it will continue to move away, indicating an unstable equilibrium. Conversely, at \( y = -2 \), the slope segments point towards it, implying that any small deviation will eventually return to this equilibrium, marking it as stable.
Stability analysis, through slope fields or other methods, is crucial in fields like ecology, physics, or economics because it helps predict how systems might respond to changes or disturbances.