Finding current through a capacitor in a series AC circuit
You didn't set up the problem correctly. Ic(t) is proportional to the derivative of Vc(t) -- the voltage across the capacitor. V(t) -- the driving voltage -- equals Vc(t) + Vr(t) where Vr(t) is the voltage across the resistor.
So to come up with the right differential equation you need to:

put in the right proportionality constant in the relation Ic(t) = (some constant)*dVc(t)/dt

relate Vr(t) to Ic(t)

Use the relationship V(t) = Vc(t) + Vr(t)