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Section 6.1 Using the Direction Field Explorer app

Subsection 6.1.1 Visualizing 1st-order Differential Equations

In differential equations we begin by studying many applications of first-order differential equations. We learn how these equations are derived from the language of the problems, and then we examine the direction fields associated with each equation to make predictions, consider equilibria and to analyze the situations.

We can use the Direction Field Explorer app to visually explore the direction fields associated with any first order differential equation that can be solved for the first derivative.

We can also visually verify the general solution to these first-order differential equations.

Direction Field Explorer image
Figure 6.1.1 Direction Field Explorer app

To do so, follow the steps below:

  1. Enter the differential equation, solved for dy/dx to see the associated direction field.
  2. Then enter the general solution to this differential equation (including the integration parameter C).
  3. Next we can verify that the solution curves generated for various values of the parameter C all fit nicely through the field no matter what value of C is used. A slider is provided to allow this parameter to be quickly and smoothly varied through a broad range of values.

The variables of the problem can be changed, and the range of values for each variable can be set to whatever is most appropriate for the problem being explored. If the range of values for one variable is much larger than for another, a non-uniform grid can be used to more easily view these situations.

Clicking at a point on the direction field causes a solution curve to be drawn in that passes through the indicated point.