RC Calculation With Charge Already On Capacitor
The answer to your question is yes. 

For simple circuits with one capacitor, usually the initial voltage on the capacitor (before anything changes) and the final value of voltage on the capacitor (after a really long time has passed), can be found by really simple circuit analysis, or by inspection of the circuit.  

the "time constant" of a circuit with only one capacitor can be found by pretending that the circuit has terminals where the capacitor is, and noting the Thevenin resistance of the circuit that is seen by the capacitor at those terminals. The time constant is then (C*Rthev). 

Now, with the three numbers Vi=initial voltage on the cap, Vf=Final voltage on the cap, tau=time constant derived from the capacitors point of view......
  The voltage on the capacitor as a function of time is V(t)= Vf-(Vf-Vi)*exp(-t/tau).