The mathematical model of a dynamic system is given as [ left{begin{array}{l} ddot{x}_{1}+2 dot{x}_{1}+frac{1}{2}left(x_{1}-x_{2} ight)=f(t) 3

The mathematical model of a dynamic system is given as

\[
\left\{\begin{array}{l}
\ddot{x}_{1}+2 \dot{x}_{1}+\frac{1}{2}\left(x_{1}-x_{2}\right)=f(t) \\
3 \ddot{x}_{2}+\dot{x}_{2}-\frac{1}{2}\left(x_{1}-x_{2}\right)=0 \end{array}\right.
\]
where \(f\) is the input and \(x_{1}\) and \(x_{2}\) are the outputs.

a. Find the appropriate transfer functions.

b. Assuming \(f(t)\) is the unit impulse, find the expressions for \(X_{1}(s)\) and \(X_{2}(s)\) by using (a).

c. Find the steady-state values of \(x_{1}\) and \(x_{2}\) by using the final-value theorem.