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. $$y^{\prime}(t)=-6 y+12$$

Short answer

Answer: The equilibrium solution is \(y(t) = 2\), and it is stable.

Step-by-step solution

Finding the equilibrium solution

To find the equilibrium solution, set the derivative equal to zeros and solve for \(y\). So, $$ y'(t) = 0 \implies -6y+12 = 0 $$ Solve for \(y\), we get: $$ y = \frac{12}{6} = 2 $$ The equilibrium solution is \(y(t) = 2\).

Analyze the stability of the equilibrium solution

To determine the stability of the equilibrium solution, we need to examine how the derivative \(y'(t)\) behaves when \(y\) is slightly above or below the equilibrium value \(y(t)=2\). If \(y > 2\): $$ y'(t) = -6y + 12 < 0 $$ This implies that when the solution is above the equilibrium, it is decreasing (\(y'(t) < 0)\). If \(y < 2\): $$ y'(t) = -6y + 12 > 0 $$ This implies that when the solution is below the equilibrium, it is increasing (\(y'(t) > 0)\). Since the solutions are decreasing when above the equilibrium (\(y(t) = 2\)) and increasing when below it, the equilibrium is stable.

Sketch the direction field

To sketch the direction field for \(t \geq 0\), we just need to indicate if the function is increasing or decreasing on either side of the equilibrium solution. - For \(y > 2\), \(y'(t) < 0\) and the solutions are decreasing. Draw the direction field with downward sloping arrows above the line \(y=2\). - For \(y < 2\), \(y'(t) > 0\) and the solutions are increasing. Draw the direction field with upward sloping arrows below the line \(y=2\). - At the equilibrium solution \(y=2\), draw a horizontal line to represent the stable equilibrium.

Key concepts

Direction Field

A direction field, also known as a slope field, is a visual representation used in differential equations to illustrate the behavior of solutions. Think of it as a map that shows the direction in which solutions to the equation are heading at any given point. To sketch a direction field for the equation given in the exercise, you need to focus on whether solutions increase or decrease around the equilibrium point.

For our example, we have the differential equation \(y'(t) = -6y + 12\). Here, the direction field will show:

• Upward sloping arrows below the equilibrium point at \(y = 2\) since the solution increases when \(y < 2\).
• Downward sloping arrows above \(y = 2\) to indicate decreasing solutions when \(y > 2\).

At \(y = 2\), the field shows horizontal lines, representing a stable equilibrium where solutions neither increase nor decrease. This visual approach helps to quickly grasp the stability nature and behavior of solutions at different points.

Stability Analysis

Stability analysis is a fundamental concept when examining differential equations. It involves studying how solutions behave as time evolves, and whether they converge to an equilibrium solution or diverge away from it.

In the given exercise, the equilibrium solution is found at \(y = 2\). The stability analysis involves examining the behavior of \(y'(t) = -6y + 12\) when \(y\) is slightly perturbed from this equilibrium point:

• If \(y > 2\), \(y'(t)\) becomes negative, indicating that solutions decrease back toward \(y = 2\). This means the system corrects itself when \(y\) is too high.
• If \(y < 2\), \(y'(t)\) is positive, and solutions increase, pushing back up toward \(y = 2\). This self-correcting behavior ensures solutions stay near the equilibrium point.

Thus, \(y = 2\) is a stable equilibrium because any deviation from this point results in behavior that drives solutions back toward equilibrium.

Differential Equation Solution

Solving a differential equation requires finding a function that satisfies the equation for all applicable values. In this exercise, you're tackling a linear first-order differential equation: \(y'(t) = -6y + 12\).

The process typically involves:

• Identifying the equilibrium solution, the state where the derivative equals zero and the system is in balance. This was found to be \(y = 2\).
• Analyzing the nature of this solution through stability analysis to determine if small deviations return to equilibria, confirming its stability.
• Providing a direction field sketch that visually maps out the general behavior of possible solutions around the equilibrium point.

These steps not only help find specific solutions but give insight into the overall dynamical behavior of the system being studied. Having a clear visual and analytical understanding can significantly aid in predicting and controlling systems modeled by differential equations.