Question

Draw the directional field associated with the differential equation, then solve the differential equation. Draw a sample solution on the directional field. $$ y^{\prime}=2 y-y^{2} $$

Short answer

Solve the differential equation and visualize it using a directional field.

Step-by-step solution

Analyze the Differential Equation

We start with the differential equation \( y' = 2y - y^2 \). Our goal in this step is to understand the behavior of the differential equation. Notice that it is a first-order separable differential equation.

Draw the Directional Field

To draw the directional field for \( y' = 2y - y^2 \), assign values to \( y \) and compute \( y' \). For example, if \( y = 0 \), then \( y' = 0 \); if \( y = 1 \), then \( y' = 1 \); if \( y = 2 \), then \( y' = 0 \). Plot these slopes at corresponding \( (x, y) \) points. Repeat several times to establish the field.

Solve the Differential Equation

To solve the differential equation, separate variables: \( \frac{1}{2y - y^2} \) dy = dx. Integrate both sides: for the left side use partial fraction decomposition, \( \frac{1}{2y - y^2} = \frac{1}{y} + \frac{1}{2-y} \). Integrating gives \( \ln |y| - \ln |2-y| = x + C \). Simplify to \( \frac{y}{2-y} = Ae^x \), where \( A = e^C \).

Add Sample Solutions to the Directional Field

Plot the solution \( \frac{y}{2-y} = Ae^x \) on the directional field for different values of \( A \). These plots represent specific solution curves and follow the arrows drawn in the directional field.

Key concepts

Directional Field

A directional field, also known as a slope field, is a graphical representation that helps us visualize the behavior of a differential equation at various points on the plane. For the differential equation \( y' = 2y - y^2 \), the directional field demonstrates the slope of the solution curve at any given \( (x, y) \) position. To create a directional field, we calculate the slope (\( y' \)) for different values of \( y \) and plot these values.

• When \( y = 0 \), \( y' = 0 \): The slope here is horizontal.
• When \( y = 1 \), \( y' = 1 \): The slope is positive, suggesting an upward tilt.
• When \( y = 2 \), \( y' = 0 \): Again, the slope is horizontal.

By repeating this for various values of \( y \), and plotting these slopes at the corresponding \( (x, y) \) points, you create a field of tiny line segments. These segments help us predict the trajectory of the potential solution curves throughout the graph.

Separable Differential Equations

Separable differential equations are a class of differential equations where the variables can be separated on opposite sides of the equation. This allows us to solve them using integration. For instance, with the equation \( y' = 2y - y^2 \), we can express it in a form that allows separation:
\[\frac{1}{2y - y^2} \, dy = dx\]
This equation is separable because we have isolated all terms involving \( y \) on one side and all terms involving \( x \) on the other.Once separated, integration can proceed on both sides. This process can sometimes be complex, requiring additional techniques such as partial fraction decomposition to proceed with integration effectively.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down complex rational expressions into simpler parts that are easier to integrate. In our example, we have the function \( \frac{1}{2y - y^2} \).
To decompose this, we rewrite it as:\[\frac{1}{2y - y^2} = \frac{1}{y} + \frac{1}{2-y}\]
This decomposition allows us to integrate each term individually:

• \( \int \frac{1}{y} \, dy = \ln |y| \)
• \( \int \frac{1}{2-y} \, dy = -\ln |2-y| \)

These simpler integrals are much easier to handle and are crucial in solving the differential equation effectively, paving the way for finding the general solution.

Solution Curves

Solution curves, also known as integral curves, represent the functions that satisfy the differential equation under given initial conditions. In the context of our equation, \( \frac{y}{2-y} = Ae^x \), various values of \( A \) produce different solution curves.
To plot these, substitute different constants \( A \) to represent specific solutions on the directional field.

• Each curve follows the direction indicated by the field's arrows.
• The constant \( A \) determines the particular solution's Location and shape.
• They depict how the system evolves over time for different initial conditions.

By plotting these on the pre-drawn directional field, you can visualize how solutions behave and interact with the overall graphic setup, greatly aiding in understanding the dynamics described by the differential equation.