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Slope Fields with Mathematica

Equations of the Form: dy/dx = g(x, y)

General Observations

The common themes that you should have noticed throughout the set of exercises are the following:

  • The slope fields of the differential equations in this class showed much more variation than those from equations in the previous two classes. Isoclines were no longer predictable vertical or horizontal lines, but could be slanted lines or in most cases curves of a quite complicated nature.

  • The exercises we did involving differential equations from this class were solvable by methods we may or may not have heard of. They fell into several groups: separable, linear, exact, and Bernoulli. In fact we have only scratched the surface—there are quite a few other types of equations of this general form.

  • We didn't encounter this in the exercises, but the vast majority of differential equations are not solvable by any analytic technique and we have to resort to approximation methods and pictures (like the slope field) in order to analyze their behavior.

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ODE Laboratories: A Sabbatical Project by Christopher A. Barker

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