Consider the solutions to differential equations as paths through direction fields. A solution to a differential equation is a function that satisfies the differential equation. 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.