How do I find the current equations for the RLC Series Circuit?
In your first equation, write \$\small  V_c= \frac{1}{C}\large \int \small i\:dt \$, then differentiate the whole of this equation (including the constant source voltage).

This gives:

$$\small 0=R \frac{di}{dt}+L\frac{d^2i}{dt^2} +\frac{1}{C}i $$

Rearranging:

$$\small \frac{d^2i}{dt^2}+ \frac{R}{L} \frac{di}{dt}+\frac{1}{LC}i=0 $$

Then the auxiliary equation is: \$\small m^2+\frac{R}{L}m+\frac{1}{LC}=0 \$, etc...