Obtaining a second order differential equation.

Question

Supply voltage of a DC motor is given by:
V(t)=Ri(t)+L (d i(t))/dt+kω(t)
Here i(t) is the current through the coil, R is the resistance, L is the inductance of coils, ω(t) is the angular speed and k is a constant.
The torque of the motor is given by,
T=J (d ω(t))/dt+Bω(t)=k.i(t) (Here J and B are constants)
Using the above two equations, obtain a second order differential equation for ω(t), which relates ω(t) and V(t).

Answer 1

There is a k in both equations.  I don't know if they are supposed to be the same.  Let's assume they are different and denote the one in the first equation by k and the one in the second equation by K.  Also denote the differential operator by D.

 

Second equation gives:

 

i=(1/K)(JDw+Bw)

 

Di=(1/K)(JD²w+BDw)

 

Substitute for i and Di in the first equation.

 

V=R[(1/K)(JDw+Bw)]+L[(1/K)(JD²w+BDw)+kw

 

V=(LJ/K)D²w+[(RJ/K)+(LB/K)]Dw+[(RB/K)+k]w

 

(LJ/K)D²w+(1/K)(RJ+LB)Dw+[(RB/K)+k]w=V

 

Check to make sure I didn't drop something along the way.