At $t=0$ capacitor is uncharged and flux of inductor is $\varphi$.

Now, using $\varphi =LI$, at $t=0$ current in circuit is

$I_0=\frac{\varphi }{L}$

Instantaneous current in circuit is $\frac{dq}{dt}$.

where, $q$ is solution of $L\frac{d^{2}q}{d{t}^2}+\frac{q}{C}=0$

or $\frac{d^{2}q}{d{t}^2}+\frac{q}{LC}=0$

At $t=0$, current in circuit is non-zero, so instantaneous current is given by 

$=I_0\cos \omega t=\frac{\varphi }{L}·\cos \omega t$