$(25$pts$)$ Consider the system given in below figure.

a$)(10$ points$)$ Derive the mathematical equations if the variables are $x_1$ and $x_2$ and the input is torque $T_m.$

b$)(15$ points$)$ Consider the values as $m=50kg,k_1=1000\frac{N}{m},k_2=500\frac{N}{m},b=20N\frac{s}{m},J=1kg.m^2,B=0\mathrm{.}5N.m.\frac{s}{r}ad,R=0\mathrm{.}3m,T_m=50Nm.$ What will be the response of the system for both are $x_1$ and $x_2?$

MATLAB functions that you may need:

step$($sys$)$ plots the response of a dynamic system model to a step input of unit amplitude.

tf$($numerator$,$ denominator$)$creates a continuous$-$time transfer function model