Problem $6(8$ pts$.)$: A circuit is excited by a single voltage source $v_s(t)=4cos(10t).$ The general solution for a particular voltage of interest $v(t)$ is:

$v(t)=4+5e^{-2t}+2sin(10t)+3cos(10t)+1\mathrm{.}5e^{-3t}cos(20t+30)$

a$)$ Determine the natural and forced components of the $v(t)$ response.

b$)$ Determine the transient and steady$-$state components of the $v(t)$ response.