Slope Fields/Euler's Solution

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Slope fields are a powerful tool for checking the validity of derivatives, anti-derivatives, and solutions to differential equations. While the graphical resolution and processing speed of a TI-83 limit the usefulness of the techniques somewhat, it is still a handy tool to have loaded in one's calculator on examination day.

A related technique, known as 'Euler's Solution' after the eighteenth-century Swiss mathematician Leonhard Euler, traces a particular solution curve from a given set of initial coordinates. By applying regression models to the data accumulated while tracing an Euler's solution, it is occasionally possible to determine an expression that will approximate Euler's curve. Again, given the platform (TI-83, that is), the likelihood of this actually yielding useful results is slim, but it's interesting to explore.

Note: This program uses the calculator's 'Y0', 'Y9', and 'Y8' graphing functions in it's calculations, and will overwrite any equations previously stored there.

Using the Program:

The following example demonstrates using the program to explore the slope field described by dY/dX = 2x.

Instructions:	Screen Shot
1) Start the program. The main program menu appears, as shown at right. The function entered as dY/dX is used to draw the slope field. The window setup can be edited via the 'Edit Setup' option.
2) The setup menu is shown at right. Values entered here are used in the program's various displays.
3) After selecting option '1' from the main menu, the expression describing the slope field is entered. Press 'Enter' when done.
4) Selecting option '3' from the main menu now draws the slope field depicted at right. This image is stored and superimposed over various Euler's or user-entered solution curves later in the program. Press 'Enter' to proceed.
5) Selecting option '4', 'Euler's Method', brings up the series of prompts shown next. The values entered here specify an Euler's curve starting at (1.5,1.25) with a step size of 0.5. Press 'Enter' after entering each value.
6) The Euler's Solution curve is traced out, with the slope field overlaid. Press 'Enter' to proceed.
7) You are now offered the option of applying one of the calculator's pre-defined regression procedures to the data generated in the Euler's Solution step. In this case we selected '1'.
8) Select one of various regression models. We selected '1', quadratic regression.
9) The solution presented by the regression model is x^2 -.05x - 1.075. The actual particular solution for the given initial conditions would have been x^2 - 1. Press 'Enter' to continue.
10) Press '5' at the main menu to draw your own solution and compare it with the slope field. Press 'Enter' to continue.
11) The solution entered in the previous step is shown, with the slope field superimposed. Press 'Enter' to return to the main menu.

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