11 |
Dynamics of Change: Differential Equations and Proportionality |
Project 11.1: Slope Fields |
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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.
y(x)=x^{2}+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)=x^{2}-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. 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. |
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