Question

Use the window \([-2,2] \times[-2,2]\) to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. \(A\) detailed direction field is not needed. $$y^{\prime}(t)=y(2-y), y(0)=1$$

Short answer

Based on the provided step-by-step solution to sketch the direction field and the solution curve, the short answer is as follows: Analyzing the given first-order autonomous differential equation, we found the equilibrium points at y = 0 and y = 2. The function is increasing between these equilibrium points and decreasing above y = 2. Using this information, we sketched a direction field in the rectangular window [-2, 2] × [-2, 2] by drawing horizontal line segments for nullclines and upward or downward sloping line segments for increasing or decreasing intervals respectively. Given the initial condition y(0) = 1, we sketched the solution curve that starts at the point (0, 1) and follows the direction field by increasing in both positive and negative time directions.

Step-by-step solution

Analyze the differential equation

Observe the given differential equation: $$y^{\prime}(t) = y(2-y)$$ We can notice that this differential equation is autonomous, meaning that it does not depend explicitly on the variable \(t\). This implies that our direction field will be consistent in the vertical direction. We should also look for possible equilibrium points and nullclines, as they give us more insights into the solution behavior. Nullclines are when \(y^{\prime}(t) = 0\), therefore: $$y(2-y) = 0 \Rightarrow y = 0 \ \textrm{or} \ y = 2$$ These two lines, \(y = 0\) and \(y = 2\), will represent the equilibrium points of our solution curve.

Find the solution behavior for different intervals

Now, we can analyze how our function behaves in the different regions separated by the nullclines. We have three intervals to check: \(y < 0\), \(0 < y < 2\), and \(2 < y\). 1. For \(y < 0\), we have \(y(2-y) > 0\), which means that the function is increasing in this region. 2. For \(0 < y < 2\), we have \(y(2-y) > 0\), also increasing. 3. For \(2 < y\), we have \(y(2-y) < 0\), which means that the function is decreasing in this region.

Sketch the direction field

Now that we have an idea of how the function behaves, we can sketch the direction field using small line segments to indicate the slope of the solution for each point in the rectangular window \([-2,2] \times[-2,2]\). 1. For the equilibrium lines \(y = 0\) and \(y = 2\), the horizontal line segments will be drawn, as \(y^{\prime}(t) = 0\). 2. Between the nullclines, draw upward sloping line segments, as the function is increasing. 3. Above the \(y = 2\) line, draw downward sloping line segments, as the function is decreasing.

Sketch the solution curve

We are given an initial condition: \(y(0) = 1\). The curve will start from the point \((0, 1)\). Since \(0 < y < 2\), we know that our solution curve should be increasing. Follow the direction field from this initial point with an increasing trajectory in both the positive and negative time directions. To summarize: 1. For \(t > 0\), the solution curve increases as it moves along the positive time axis. 2. For \(t < 0\), the solution curve also increases as it moves along the negative time axis. Sketch the solution curve accordingly.

Key concepts

Direction Field

In differential equations, a direction field, also known as a slope field, is a visual representation that helps us understand how solutions behave. It consists of small line segments or arrows drawn on a graph, where each line represents the slope of the solution curve at that point. A direction field can give us insights into the nature of solutions without finding the exact solution analytically.

For our equation, the direction field tells us the direction in which the solution curve should move. We observe that between the lines of equilibrium, the arrows point upwards, indicating an increasing solution. Beyond these lines, the arrows change direction according to the behavior of the equation. This visual tool is invaluable for sketching rough solution paths and understanding how the differential equation behaves over a specified range.

Autonomous Equation

Autonomous equations are a special type of differential equation. They have the form \( y' = f(y) \), where the rate of change depends only on the function itself, not on the independent variable, such as time \( t \). This independence implies that the direction field will look the same at any horizontal slice of the graph.

In our case, \( y' = y(2-y) \) is autonomous. This means the slope of the solution curve at any given \( y \) value remains the same for all \( t \). What makes autonomous equations valuable is their simplicity and regularity. The direction field will not shift as \( t \) progresses, which allows us to easily predict and understand the long-term behavior of the solution.

Equilibrium Points

Equilibrium points are where the solution to the differential equation remains constant, meaning the derivative is zero. For our example, we find these points by setting \( y' = 0 \).

Solving \( y(2-y) = 0 \), we get equilibrium points at \( y = 0 \) and \( y = 2 \). These points hold crucial information:

• For \( y = 0 \), the solution neither decreases nor increases, creating a horizontal direction field line.
• For \( y = 2 \), the same horizontal effect occurs.

Understanding where these equilibrium points occur in the field helps predict solution behavior and stability. They effectively define the regions where the function will exhibit increasing or decreasing trends.

Solution Curve

The solution curve represents the path that a particular solution takes through the direction field based on a given initial condition. It forms the trajectory that follows the arrows or line segments of the field.

Given the initial condition \( y(0) = 1 \), the solution curve starts at the point \( (0, 1) \). According to the direction field constructed:

• For \( 0 < y < 2 \), this trajectory follows an upward slant due to increasing direction field lines.
• The curve will continue to rise because it remains below the equilibrium at \( y = 2 \).

The solution curve emphasizes the concept that initial conditions dictate how solutions move, while the direction field provides a backdrop illustrating possible paths.