Using a direction field, we can see many possibile solutions. Imagine a river with a current given by the direction field. If a leaf were to fall into the river it would be swept along a path determined by those currents. The curve that the leaf sweeps out corresponds to a solution of the differential equation. However, if the leaf were to have landed in a slightly different location in the river, the path it takes may be quite different.
Lets choose the origin.
Its also possible to view an entire family of solutions at once by using Maples ability to create a set of different points to consider. Each point will specify a different solution. The set of all of these solutions form a family of solutions. The problems above had simple answers because each differential equation could be integrated to get a solution. In this section we will do the same thing - plot a direction field and various solutions which flow as trajectories in the direction field.