This can also be called time independent . Since the slopes of the lines in the field only depend upon the variable p , all of the segments on a given horizontal line have the same slope. This is what makes autonomous equations and their solutions so special. It implies that every solution curve can be obtained from a given curve by sliding its graph left and right. See the picture.
All curves are horizontal translates of each other.
This will be referred to as a derivative profile . To clarify the relationship between the derivative profile and the slope field, the profile is sketched below with a vertical p axis. Their are two special solution curves in the example above. These solutions are called equilibrium solutions . The slope field indicates that as t increases, every solution either moves towards or moves away from an equilibrium solution.