Question

Potential difference across each capacitor at that instant.

Answer 1

$q_1=8C{V}_0$: $q_2=C{V}_0$
$q_1+q_2=9C{V}_0$
In the absence of inductor, this $9C{V}_0$ will distribute as $6C{V}_0$ in $2C$ and $3C{V}_0$ in $C$. Thus, mean position of $q_1$ is $6C{V}_0$ and mean position of $q_2$ is $3C{V}_0$.
At $t=0$, $q_1$ is $2C{V}_0$ more than its mean position and $q_2$ is $2C{V}_0$ less.
Thus, $q_0=2C{V}_0$
$C_{net}=\frac{2C}{3}$
$\therefore$ $\omega =\frac{1}{\sqrt{L{C}_{net}}}=\sqrt{\frac{3}{2LC}}$
$V_1=\frac{6C{V}_0}{2C}=3V_0$
and $V_2=\frac{3C{V}_0}{C}$
$=3V_0$