The circuit equations for the $L C$ circuit are: $$ L \frac{\mathrm{d} i}{\mathrm{d} t}+\frac{1}{C} \int i \mathrm{d} t=0 $$

$$\therefore \quad L \frac{\mathrm{d}^{2} i}{\mathrm{d} t^{2}}+\frac{1}{C} i(t)=0$$

Taking laplace transform of the above equation, $\left(S^{2} L+\frac{1}{C}\right) I(s)=0$

$$\begin{array}{ll}{\therefore \qquad i(t)} & {=E_{\mathrm{dc}} \sqrt{\frac{c}{L}} \sin \omega_{0} t} \\ {\text { where }} …