11 Dynamics of Change: Differential Equations and Proportionality

Project 11.1: Slope Fields

Introduction

This chapter presented Euler's method for approximating a point on the graph of a particular solution to a differential equation. It is possible to go one step further and visualize the shapes of the graphs of particular solutions to a differential equation using a picture called a slope field.

A differential equation tells us the slope of the graph of a solution function at a particular point. If a grid is placed on the xy-plane and a short line segment whose slope is determined by the differential equation is drawn at each point on the grid, then the result is a slope field for the differential equation. For instance, consider the differential equation . At the point (1,1), the slope of a solution to this equation is 2x=2(1)=2 and at (-0.5,2), the slope is 2x=2(0.5)=-1. Using the differential equation to determine the slopes at the points on a grid and sketching short line segments with those slopes at the appropriate points gives the slope field shown in Figure 11.7. (This construction is a tedious process and is usually done with a computer program.)

Particular solutions for the differential equation can be sketched by following the line segments in such a way that the solution curves are tangent to each of the segments they meet. Figure 11.8 shows the graphs of several particular solutions for.

 FIGURE 11.7 FIGURE 11.8

You should recognize the curves in Figure 11.8 as graphs of solutions of the form

y(x)=x2+C which we know is the general solution of. If you are given an initial condition, then you can sketch a particular solution by following the direction indicated by the line segments from the starting point. The particular solution in Figure 11.8 with initial condition (-2,3) is shown in blue. In this case, the particular solution is y(x)=x2-1. The other particular solutions shown correspond to C = -3, -2, 0, 1, 2, 3.

While this example of a slope field is for a differential equation with a known solution, solutions are particularly helpful when graphing solutions for a differential equation for which we do not know a formula for the solutions.

Figures 11.9 through 11.16 show 8 different slope fields.

1. Describe the information about the differential equation (that is, the behavior of the slopes) presented by each slope field graph.

2. Reproduce the slope field graphs and, on each one, sketch four particular solutions. Identify each particular solution you sketch with an initial condition.

3. The differential equations whose slope field graphs are shown are listed among the following. Identify which differential equations correspond to each slope field. Explain your reasoning.

 (1) (2) (3) (4) (5) (6) (7) (8), k > 0 (9) (10)

 FIGURE 11.9 FIGURE 11.10 FIGURE 11.11 FIGURE 11.12 FIGURE 11.13 FIGURE 11.14 FIGURE 11.15 FIGURE 11.16

Reporting

1. Prepare a written report of your work. Include all graphs, appropriately labeled. Include discussions of each of the foregoing tasks.

2. (Optional) Report your results on a poster. Your poster should be neat, clearly labeled, and easy to read from 3 feet away.