Question

Consider the following differential equations. A detailed direction field is not needed. a. Find the solutions that are constant, for all \(t \geq 0\) (the equilibrium solutions). b. In what regions are solutions increasing? Decreasing? c. Which initial conditions \(y(0)=\) A lead to solutions that are increasing in time? Decreasing? d. Sketch the direction field and verify that it is consistent with parts \((a)-(c)\). $$y^{\prime}(t)=(y-2)(y+1)$$

Short answer

Question: Find the equilibrium solutions, determine where solutions are increasing or decreasing, identify initial conditions that lead to increasing or decreasing solutions, and verify the results based on the given first-order differential equation: $$y^{\prime}(t)=(y-2)(y+1)$$ Answer: a) The equilibrium solutions are \(y=-1\) and \(y=2\). b) Solutions are increasing when \(y<-1\) or \(y>2\), and decreasing when \(-12\), the solution will be increasing, and if \(-1<A<2\), the solution will be decreasing. d) The direction field should be consistent with the calculations for equilibrium solutions and the regions where solutions are increasing or decreasing.

Step-by-step solution

Identify the given differential equation

The given first-order differential equation is: $$y^{\prime}(t)=(y-2)(y+1)$$

Find the equilibrium solutions

To find the equilibrium solutions, we need to find the constant solutions. Equilibrium solutions occur when the derivative of \(y\) with respect to \(t\) is zero. So, we must have: $$0 = (y-2)(y+1)$$ We can find the constant solutions by solving this equation for \(y\).

Solve the equation for equilibrium solutions

To find the values of \(y\) for which \((y-2)(y+1)=0\), we set each factor equal to zero: $$y-2=0 \Rightarrow y=2$$ $$y+1=0 \Rightarrow y=-1$$ These are the equilibrium solutions.

Determine where solutions are increasing or decreasing

We can analyze the sign of \(y'(t)\) to determine where the solutions are increasing or decreasing: - If \(y'(t)>0\), the solution is increasing. - If \(y'(t)<0\), the solution is decreasing. Let's find the regions for which \(y'(t)>0\) and \(y'(t)<0\).

Find the regions for which \(y'(t)\) changes sign

We are looking for the regions in which the product \((y-2)(y+1)\) is positive or negative: - If \(y<-1\), both factors are negative, and their product is positive. - If \(-12\), both factors are positive, and their product is positive. So, when \(y<-1\) or \(y>2\), the solution is increasing, and when \(-1<y<2\), the solution is decreasing.

Identify initial conditions that lead to increasing or decreasing solutions

Based on our analysis in Steps 4 and 5: - If the initial condition \(y(0)=A<-1\) or \(y(0)=A>2\), the solution will be increasing. - If the initial condition \(y(0)=A>-1\) and \(y(0)=A<2\), the solution will be decreasing.

Sketch the direction field and verify the results

To sketch the direction field, we can use software tools or graph by hand. In either case, we should see that the direction field is consistent with the equilibrium solutions \(y=-1\) and \(y=2\), and with the regions where the solution is increasing or decreasing, as determined in Steps 4, 5, and 6. To summarize: a) The equilibrium solutions are \(y=-1\) and \(y=2\). b) Solutions are increasing when \(y<-1\) or \(y>2\), and decreasing when \(-12\), the solution will be increasing, and if \(-1<A<2\), the solution will be decreasing. d) The direction field should be consistent with the calculations for equilibrium solutions and the regions where solutions are increasing or decreasing.

Key concepts

Equilibrium Solutions

In the context of differential equations, equilibrium solutions represent the values where the change in the function stops, meaning the derivative equals zero. These are constant solutions that do not change over time. For the equation given, we have:
\[ y' = (y-2)(y+1) = 0 \]This equation implies that the product of the two factors must be zero. Therefore, the solutions to this are where

• \(y - 2 = 0\) leading to \(y = 2\), and
• \(y + 1 = 0\) leading to \(y = -1\).

These are the equilibrium solutions, meaning if the function starts at either of these values, it remains unchanged over time.

Direction Field

The direction field, or slope field, provides a visual representation of a differential equation. It consists of tiny line segments or arrows on a graph that depicts the slope of the solution at various points. For our equation \(y' = (y-2)(y+1)\), the slope field shows how the function behaves for different initial values of \(y\).
In areas where the slope is zero, we see the equilibrium solutions, \(y = -1\) and \(y = 2\). These lines appear flat on the graph. In regions where \(y > 2\) or \(y < -1\), the slope is positive, indicating the solutions are increasing. Conversely, where \(-1 < y < 2\), the slope is negative, showing the solutions are decreasing. This visual aid helps us understand how the behavior shifts around these critical points.

Increasing and Decreasing Functions

The terms "increasing" and "decreasing" describe how functions change as you move from left to right across the graph. For the differential equation \(y' = (y-2)(y+1)\), the behavior of the function changes in different regions:

• For \(y > 2\) or \(y < -1\), \(y'(t)>0\) making the functions increase.
• For \( -1 < y < 2\), \(y'(t)<0\), thus the functions decrease.

By checking the signs of the product \((y-2)(y+1)\), we can determine whether the slope is positive or negative, allowing us to identify which direction the function is moving. An understanding of these regions is key to predicting the behavior of solutions over time.

Initial Conditions

Initial conditions are particular values assigned to solutions at the starting point—often when \(t = 0\). These conditions significantly influence the future behavior of the solutions to a differential equation. For our equation:
- If \(y(0) = A > 2\) or \(y(0) = A < -1\), the solution will increase over time.- If \(-1 < y(0) = A < 2\), the solution will decrease over time.
Thus, knowing the initial value of \(y\) helps forecast whether the function will rise or fall as time progresses. Analyzing these starting points in the context of equilibrium solutions and the direction field helps create a complete picture of the dynamic behavior of the system. This understanding is vital for predicting long-term trends and behaviors of solutions based on their initial conditions.