Transfer Function of DC Servo Motor

Direct current maybe - but how can the motor speed be controlled if the DC does not change in value?

the dc motor speed can be controlled by applied voltage,or armature current.the dc motor may run at constant speed after balance in between applied voltage =back emf+ armature resistive drop.but in finding transfer function we are assuming that applied voltage & armature currents are functions of time why?

To run at constant speed with constant applied voltage implies a constant load. If the load is constant, just measure the motor speed and adjust the applied voltage to achieve your required speed. Indeed, you don't need to know the inductance. Neither do you need a servo motor + drive.

A motor is an electro-mechanical device. You cannot describe the transfer function from just the electrical side of the system. You must include the mechanics to get a valid transfer function. (Ignoring for now the need for commutation.) The mechanics will develop in the electric circuit a back EMF (voltage) source in the circuit where the angular velocity of the rotor will be proportional to the EMF. The current through the circuit will be proportional to the torque developed on the rotor and any mechanical load attached to the shaft. The torque will be proportional to the angular acceleration of the mechanical loads. So the back EMF will change with time because it is a first order derivative of the current in the circuit. When one includes the real world conditions of drag from friction, resistive losses, cooling wind losses and core losses this turns into a second order differential equation. Adding the directional effect of gravity if one tries to lift a mass makes for a conditional pair of differential equations depending on when gravity assists or fights the desired motion. So even though you have just one external DC source, the time varying back EMF source will change the voltage across your inductor.

Now finish your homework.

Did I mention the non-linear effects of stiction, commutation, gear lash or even slippage? Never mind, that's enough for now.

A motor behaves differently depending on what it is attached to it. Its behaviour will be different when attached to a large fan compared to when it is attached to a large flywheel for example.

Consider the mechanical arrangements before proceeding further.