Slope Fields with Mathematica - Exercise 1 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 y-value 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.