Please solve $3\mathrm{.}41$ as the textbook says. Additionally, please solve the next three parts written here.

$3\mathrm{.}41$ Figure P$3\mathrm{.}41$ illustrates a pendulum with a base that moves. The base acceleration is $a(t).$ Derive the equation of motion in terms of $$ with $a(t)$ as the input. Neglect the mass of the rod.

For problem $3\mathrm{.}41,$ in addition to the requirements for this problem stated in the textbook, also do the following:

i$)$ Find the equilibrium positions of the pendulum.

ii$)$ Linearize the equation of motion about theta$=0,$ using the Taylor Series. You must show all steps in this linearization process, including the Taylor Series.

iii$)$ If a$($t$)$ is constant, and a$>0,$ use the linearized equation of motion to determine the stability of the unforced system, about theta$=0,$You must determine stability from the characteristic roots, and you must show the characteristic equation, and the steps used to find the characteristic roots.