
Slope Fields with Mathematica
Equations of the Form: dy/dx = g(x)
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
vertical isoclines. This makes perfect sense when you think about it.
As we discussed in the general introduction to slope fields, the equation
itself basically is a disguised form of:
slope = g(x)
Clearly then, the slope at any point only depends on x, and never on
y. If you move up or down to various points on a vertical line then only
their yvalue changes, which leads to no change in the slope. Hence,
vertical lines are isoclines.
All of the differential equations in this class are solvable by direct
integration, (assuming that the integral is analytically possible.) Indeed,
if this were the only type of differential equation we ever encountered there
would be no need for a course dedicated to studying these monsters—we could have
solved them all back in your calculus course. Unfortunately most of the really
interesting and useful differential equations aren't in this category.
It looks like we've exhausted the main points of interest concerning
equations of this type. Let's now go back to the main exercise menu.

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