Problem 10 - Differential Equations and Linear Algebra

Question

Sketch the slope field and some representative solution curves for the given differential equation. $$y^{\prime}=y(2-y)(1-y)$$

Step-by-Step Solution

Answer

Given the differential equation \(y'=y(2-y)(1-y)\), we first find the slope function \(f(y) = y(2-y)(1-y)\). We then calculate values of the slope function for some representative points, such as y = 0, 0.5, 1, and 2. Next, we plot the slope field on the xy-plane using these values, creating short line segments at each point with the corresponding slope. Finally, we sketch representative solution curves, called integral curves, that follow the slope field. These integral curves represent the general solutions to the given differential equation.

Step 1: Find the slope function of the given differential equation

Given the differential equation, \(y'=y(2-y)(1-y)\) The slope function associated with this differential equation is: \(f(y) = y(2-y)(1-y)\)

Step 2: Calculate values of the slope function

For a few representative points, calculate the value of the slope function \(f(y) = y(2-y)(1-y)\). This will help to create the slope field. For example, let's calculate the slope at y = 0, 0.5, 1 and 2. For y = 0: \(f(0) = 0(2-0)(1-0) = 0\) For y = 0.5: \(f(0.5) = 0.5(2-0.5)(1-0.5) = 0.375\) For y = 1: \(f(1) = 1(2-1)(1-1) = 0\) For y = 2: \(f(2) = 2(2-2)(1-2) = 0\)

Step 3: Plot the slope field

Plot the slope field on the xy-plane using the calculated values in step 2. Use short line segments with positive, negative, and zero slopes at the representative points. For example, for the points we calculated in step 2: - At y = 0, the slope is 0, so plot horizontal line segments along the y=0 line, since the slope is 0 everywhere along this line. - At y = 0.5, the slope is 0.375, so plot small line segments with positive slope along the y=0.5 line, corresponding to this slope value. - At y = 1, the slope is 0, so plot horizontal line segments along the y=1 line, since the slope is 0 everywhere along this line. - At y = 2, the slope is 0, so plot horizontal line segments along the y=2 line since the slope is 0 everywhere along this line. Repeat this process for other values of y to create a more complete slope field.

Step 4: Plot representative solution curves

Now that the slope field is plotted, we can now sketch some representative solution curves that follow the given slope field. These curves are commonly known as integral curves, and they represent the general solutions to the differential equation. Note that these curves may be rounded or asymptote between the known points based on the behaviors of the slope field. Plot several representative curves on the same graph with the slope field in order to visualize different possible solutions. The end result should be a graph with the slope field and representative integral curves, which provides a visual understanding of the general solutions to the given differential equation.

Key Concepts

Slope Field

A slope field, also known as a direction field, provides a visual representation of a differential equation without solving it explicitly. By plotting many short line segments across a plane, we can convey the direction in which the solution of the differential equation progresses.

To create the slope field for the differential equation \( y' = y(2-y)(1-y) \), we calculate the slope at various points. The slope function \( f(y) = y(2-y)(1-y) \) describes the slope of the tangent to the solution curve at any point \( (x, y) \).

For example:

• At \( y = 0 \), the slope \( f(0) = 0 \).
• At \( y = 0.5 \), the slope \( f(0.5) = 0.375 \).
• At \( y = 1 \), the slope \( f(1) = 0 \).
• At \( y = 2 \), the slope \( f(2) = 0 \).

These values help in laying out the slope field, allowing us to draw horizontal, inclined, or vertical line segments depending on the slope value. This comprehensive view enables students to grasp possible behaviors of solution curves at a glance.

Solution Curves

Solution curves represent the actual paths that solutions to a differential equation follow on a graph. When we look at a slope field, these solution curves align with the directional cues provided by the short line segments.

In the context of the differential equation \( y' = y(2-y)(1-y) \), solution curves are sketches that show the trajectory of a system over time. These curves aren't computed by an exact known formula but are guided by the slope field, reflecting how the solution might evolve.

When you trace a solution curve on the slope field, it should smoothly connect the segments, ensuring it follows the indicated slopes. Visualizing these curves is valuable because it shows multiple potential states or outcomes depending on different initial conditions, helping to understand the width of behavior the system might display.

Integral Curves

Integral curves are specific solution curves that directly solve a differential equation. These curves are significant because each represents a family of solutions for a particular initial condition in differential equations.

For the differential equation \( y' = y(2-y)(1-y) \), an integral curve is one that follows the precise path dictated by this equation. While plotting these curves, you can follow the traced slopes of the slope field. Integral curves make the abstract solutions to differential equations concrete by providing exact trajectories which these solutions take.

To identify this in practice, when sketching integral curves on a slope field, you see them weave through the field, touching line segments at matching slopes. This visual aid can greatly help solidify the understanding of how functions of solutions might behave.

Slope Function

The slope function is fundamental in working with differential equations as it defines the slope or direction of the tangent line to the solution curve at each point \( (x, y) \). This understanding is central to plotting the slope field.

For the given equation \( y' = y(2-y)(1-y) \), the slope function \( f(y) = y(2-y)(1-y) \) allows us to determine the angle of slope at each chosen point. This is crucial because:

• The slope indicates whether and how sharply you rise or fall at that point.
• Identifying zero values of the slope function (e.g. points where the function cancels out) reveals horizontal asymptotes of solution curves.

Understanding the slope function enlightens how the solution curves pivot, rise steeply, or level off. Thus, calculating and analyzing the slope function is a pivotal step when dealing with differential equations and their graphs.