Final answer:
The differential equation for the velocity v of a skydiver with mass m falling under gravity and experiencing air resistance proportional to velocity is dv/dt = g - (k/m)v. The mass function m(t) is not directly provided, but one could derive it from an additional differential equation if the mass were to change over time.
Step-by-step explanation:
To express the fact as a differential equation for the mass m(t), we use the given situation where the force of air resistance, FD, is proportional to velocity, v, with a constant of proportionality, k. The differential equation for the velocity, v, of the falling skydiver can be written as dv/dt = g - (k/m)v, where g is the acceleration due to gravity, and m is the mass of the skydiver.
To include the initial mass m0, we'd solve this differential equation with the initial condition v(0) = 0. Integrating, we find the solution v(t) = (mg/k)(1 - e-kt/m), which describes the velocity of the skydiver at any time t after opening the parachute.
The expression for the mass m(t) as a function of time isn't directly given in the problem, but if we were looking at a situation where the mass changes over time, we would also need an initial condition such as m(0) = m0 and a differential equation that describes how mass changes with respect to time.