Problem Consider the vibrating system described by the equations:

$9x_1^{}+27x_1-3x_2=0$

$x_2^{}-3x_1+3x_2=F(t)$

with the initial conditions $x_0=0$ and $x_0^{}=\frac{1}{\sqrt[\phantom{2}]{2}}\left[\begin{array}{c}\frac{1}{3}\\ -1\end{array}\right]\frac{m}{s}$

a$)$ Calculate the response of the system if $F(t)=\sqrt[\phantom{2}]{2}cos2tN$

b$)$ Calculate the response of the system if $F(t)=\sqrt[\phantom{2}]{2}(t)N$ where $(t)$ is the Dirac impulse function.

c$)$ Calculate the response of the system if $F(t)=\sqrt[\phantom{2}]{2}(t)+\sqrt[\phantom{2}]{2}cos2tN$