Slope Fields with Mathematica - Exercise 2 General Observations

Equations of the Form:Ìýdy/dxÌý=Ìýg(y)

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 haveÌýhorizontal isoclines. This makes sense when you think about it. The equation itself basically is another way of saying:

    slopeÌý=Ìýg(y)

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

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

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