Since there will be no current in the upper-right
R resistor, all you have is two cases: (1) At near DC input, vout=vin; and, (2) at near daylight input, vout=vinRc/(R+Rc). So it must be that [1 + R/Rc][d/dt + 1/C/(R+Rc)]Vout=[d/dt + 1/C/Rc]Vin. *(Ignoring the initial value for C, for now.)