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Since this is homework, I'm not just going to give you the answer.

When all else fails, go back to first principles.  That's what handy shortcuts, like using impedance, were derived from.

In this case, you'll end up with a system of differential equations.  You have already written the equation for the capacitor current as a function of its voltage.  Note that the capacitor and resistor currents are the same.  You should be able to write the equation for the resistor current.

The voltage across the two components are also related, in that their sum must equal the driving voltage.

After you write the various individual equations, you solve the set of equations to get the capacitor voltage as a function of the driving voltage.  Note that this assumes you've already had your differential equations course.