The input-output differential equation of an electrical circuit whose input is a voltage \(v_{i}(t)\) and the output is a voltage \(v_{o}(t)\), is given by

\[C \ddot{v}_{o}+\left(\frac{1}{R_{1}}+\frac{1}{R_{2}}ight) \dot{v}_{o}+\frac{1}{L} v_{o}=\frac{1}{R_{1}} \dot{v}_{i} .\]

(a) Find the expressions for the damping ratio \(\xi\), and the undamped natural frequency \(\omega_{n}\).

(b) Given the information: \(R_{1}=1 \Omega, R_{2}=\frac{1}{2} \Omega, C=1 F, L=\frac{1}{2} H, v_{i}(t)=2 t\) and assuming zero initial conditions, use Laplace transforms to solve the differential equation, i.e., obtain the system response.

(c) Plot the system response.

(d) Illustrate that the initial and final values of the response are the same as those obtained by using the IVT and FVT.