Problem $1(50$ pts$)$: For the RLC series circuit shown below, determine the followings:

a$)$ The second$-$order equation for current $i$

b$)$ the characteristic equation for the circuit

c$)$ the roots and type of natural response

d$)$ the total response for current $i$

I. $Ci(0)=2$A and $v(0)=5V$ Given: $R=2,C=2F,L=2H,v_s=10V$