$(20$ pts$)$ Use the circuit in Fig. $1$ to answer the following questions.

Figure $1$: Schematic for Problem $1.$

a$.$ Write the time$-$domain voltage$-$current relationship for each of the basic electrical components $($resistor$,$ capacitor, and inductor$)$ in Fig. $1.$

b$.$ Derive an expression for $I(t)($the current through $R_2)$ as function of $V_s,R_1,R_2,C,L,$ and derivatives of I.

c$.$ Find an expression for the steady state response when $V_s=5V$ DC$.$