Figure $9$: A motor rotates an angle $$ with a moment of inertia $J.$

$42.$ The rotation of an electric motor can be modelled by the ODE

$J\frac{d^2}{(d)t^2}+b\frac{(d)}{(d)t}+k=Q$

where $J$ is the moment of inertia $($see Fig. $9),b$ is the damping coefficient, $k$ is the torsional stiffness constant, and $Q$ is the driving torque.

$($i$)$ Assuming $Q=1,$ what is the transfer function for the ODE?

$($ii$)$ Setting $J=1,b=2,k=1$ and $Q=k,$ show that

$=k[1-(1+t)e^{-t}]$

is the solution to the preceding ODE.