Applying KVL on RC circuit to get the natural reponse v_c(t>0)
The current through the capacitor is \$+C\frac{dV_C}{dt}\$ clockwise. Therefore if you go anti-clockwise, like what you did, the current in the circuit would be \$-C\frac{dV_C}{dt}\$. Replace this in your third equation and you will come up with the right equation.