Solution 2.9.6.10 i(t) C v C (0)L i L (0) Figure 1: Tank Circuit For the circuit in Figure 1 applying Kircho 's voltage law yields L d dt i(t)+ 1 C Z t 0 i(t)dt+ v c (0) = 0: Applying the Laplace transform then yields V (s)=+LsI(s)+ I(s) Cs + v c (0) s =0;; or s 2 L +(1=C) s ! I(s)= ;v c (0) s ;; or nally I(s)= ;v c (0)=L (s;j= p LC)(s+ j=sqrtLC) = M s;j= p LC + M  s + j= p LC : Then M = " (s;j= p LC)(;v c (0)=L) (s;j= p LC)(s + j= p LC) # s=j= p LC = (v c (0)=L) 6  (2= p LC) 6 =2 = v c (0)( q C=L=2) 6 ;=2 = v c (0)( q C=L=2) 6 =2 Then, i(t)=v c (0) q C=Lcos(t= p LC + =2): 1