We still begin with Newon’s second law, but now we assume that the forces in the object come both from gravity and from air resistance. The gravitational force is still given by For air resistance, we assume the force is related to the velocity of the object. A simple way to describe this assumption might be where is a proportionality constant and is a positive real number. The value depends on various factors such as the density of the object, surface area of the object, and density of the air. The value affects how changes in the velocity affect the force. Taken together, a function of the form is often called a power law. The differential equation for the velocity is given by
(Notice that the force from air resistance opposes motion, and points in the opposite direction as the force from gravity.) This differential equation is separable, and can be written in the separated form
For arbitrary positive the integration is difficult, making this problem hard to solve analytically. In the case that the differential equation becomes linear, and is easy to solve either using either separation of variables or integrating factor techniques. We assume and proceed with an integrating factor so we can continue practicing the process. Writing
we identify the integrating factor