Differentials/ second order/ electrical vibration

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Homework Statement


A simple series circuit has an inductor of 1 henry, a capacitor of 10^-6 farads, and a resistor of 1000 ohms. The initial charge on the capacitor is zero. If a 12V battery is connected is connected to the circuit, and the circuit is closed at t=0, find the charge on the capacitor 1 second later and the steady state charge.

From this I know initial current =0


The Attempt at a Solution


Q''+1000Q'+10^6Q=0
Q(t)=C1(e^(-500t))cos(500sqrt(3)t)+C2(e^(-500t))sin(500sqrt(3)t)
I got that far. now if I used Q(0)=0 and Q'(0)=0 I get C1=C2=0 which I think is wrong. Can someone help? Thank you in advance.
 

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