# Slope Fields with Mathematica

## Autonomous Equations

### General Observations

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

• The slope fields of all of the differential equations in this class have horizontal isoclines. This makes sense when you think about it. The equation itself basically is another way of saying:

slope = g(y)

Hence, the slope at any point only depends on y, and never on x. If you move left or right to various points on a horizontal line then only their x-value changes, not their y-value, which leads to no change in the slope. Hence, horizontal lines are isoclines.

• All of the differential equations in this class are solvable by direct integration, (assuming that the integral is analytically possible.)

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ODE Laboratories: A Sabbatical Project by Christopher A. Barker

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