Motor will have a torque vs speed curve. Load will have a torque vs speed curve. Static operation will be determined by the intersection as you probably know.
If you want to assume that the load torque is given as a function varying as a function of time and speed, than you can numerically solve the speed of the system (as function of time) if you have the rotating inertia. It amounts to an F=MA type problem except d^2/dt^2(w) plays the role of A, rotational inertia plays the role of M, and net torque (Tmotor-Tload) plays the role of F.
