How to solve this RLC circuit for when the switch is closed?
The best way would be to do it in the Laplace domain and then convert it back to the time domain. 

Clearly the current supplied by the source is equal to the inductor current. Find the equivalent impedance and divide it by the supply voltage to get the current equation.

The equivalent impedance comes out to be - 

The current equation then comes out to be - 

Now you can take the inverse laplace of this to find the time domain expression.

Method 2 would be, as it has been pointed out in the earlier comments, is to find out the damping ratio and determine the nature of the system, but that would of course require the component values. I have provided you a general method to solve it.