I NEED HELP ON A$,$B$,$ AND C PLEASE AND THANK YOU!

Problem $4(20$pts$)$

Consider the $2-$degree$-$of$-$freedom spring mass pendulum shown below $($All motion is in the plane of the

picture shown$).$ The nonlinear equations of motion are given by

$(1)x^{}+\frac{k}{m}(x-L)-$gcos$-x{}^{}{?}^2=0$

$(2)x^2{}^{}+$gxsin$+2{}^{}x{x}^{}=0$

where $L$ is the original spring length.

Linearize the equations of motion for this system. Let the reference condition be the equilibrium $($no

motion$)$ state for the pendulum mass. In particular

$($a$)$ Define a set of perturbation variables

$($b$)$ Substitute the results of Part $($a$)$ into the equations of motion

$($c$)$ Expand the equations and discard appropriate terms $($show the terms that are to be discarded$)$