RLC initial conditions
What is giong on here? What am I doing wrong?


The first thing I see is here,


  $$ v = \frac{dI}{dt} ~~ \rightarrow ~~ \frac{dI(0^+)}{dt} = \frac{1}{L}V(0^+) =- \frac{12}{L}$$


You assumed the voltage across the inductor is equal to the voltage across the capacitor. But given 6 A flowing around the circuit, the voltage across the resistor is not zero and is probably not small enough to be neglected. 

You should use KVL, considering the drop across both the capacitor and the resistor, to find the voltage across the inductor. Which will give you the rate of change in the current.