Integration - direction fields

For any point in the cartesian plane, the value of  can be represented by a direction line.

When this is done for all possible points, we have a direction field. Drawing some of the direction lines gives the following picture of the direction field.

Suppose:

• person A starts at the point (^-1,^-1) and follows the direction field
• person B starts at the point (0,^-1) and follows the direction field.

The starting points are called the initial conditions. Each path is a solution of the equation .

The solutions of the equation  consist of an infinite family of "parallel" curves. If we know one solution, then every other solution can be obtained by moving the graph up or down.
 

To find the actual solutions or curves for A and B, we need to find the values of c.
 

Direction fields can be investigated using:

• direction field java applet
• DFIELD 2001.2 more powerful applet - John Polking
• the program Slopes 1.17 which can be downloaded from the University of Arizona