Suppose we have a system with a constant inflow of a solute and a constant volume outflow, although not necessarily constant concentration. The change in volume per unit of time of the system is then
$\frac{dV_S}{dt}=RV_{in}-RV_{out} \; .$
This one can be solved easily giving
$V_S(t)=V_S(0)+(RV_{in}-RV_{out})t \; .$
For the mass of solvant in the system, we have the equation
$\frac{dM_S}{dt}=RM_{in}-\frac{dM_{out}}{dt} \; .$
Of course, the rate of outflow of mass of solvant will depend on the concentration of the solute in the system and the rate of outflow so that
$\frac{dM_{out}}{dt}=RV_{out} \cdot C_S \; ,$
leading to the mass balance equation
$\frac{dM_S}{dt}=RM_{in}-RV_{out} \cdot C_S \; .$
The concentration of solute in the system is then $C_S=M_S/V_S$ which changes in time as
$\frac{dC_S}{dt}=\frac{d}{dt}\left(\frac{M_S}{V_S}\right)= \frac{1}{V_S}\frac{dM_S}{dt}-\frac{M_S}{V_S^2}\frac{dV_S}{dt} \; .$
Combining our equations, we get
$\frac{dC_S}{dt} = \frac{1}{V_S}\left(\frac{dM_S}{dt} -C_S\frac{dV_S}{dt}\right) $
$\frac{dC_S}{dt} = \frac{1}{V_S}\left(RM_{in}-RV_{out} \cdot C_S-C_S(RV_{in}-RV_{out})\right) $
$\frac{dC_S}{dt} = \frac{1}{V_S}\left(RM_{in}-C_S\cdot RV_{in}\right) $
Remembering our solution for the volume
$\frac{dC_S}{dt} = \frac{RM_{in}-C_S\cdot RV_{in}}{V_S(0)+(RV_{in}-RV_{out})t} $
which can be solved by separation of variables
$\frac{dC_S}{RM_{in}-C_S\cdot RV_{in}} = \frac{dt}{V_S(0)+(RV_{in}-RV_{out})t} $
Integrating gives
$\frac{-1}{RV_{in}}\log(RM_{in}-C_S\cdot RV_{in}) = \frac{1}{(RV_{in}-RV_{out})}\log(V_S(0)+(RV_{in}-RV_{out})t) + K $
where we introduced some integration constant $K$ which we'll specify later. Working further out
$RM_{in}-C_S\cdot RV_{in} = A (V_S(0)+(RV_{in}-RV_{out})t)^{\frac{RV_{in}}{(RV_{out}-RV_{in})}} $
in which $\exp(K)=A$. The constant $A$ should be chosen in such a way that
$RM_{in}-C_S(0)\cdot RV_{in} = A (V_S(0))^{\frac{RV_{in}}{(RV_{out}-RV_{in})}} \; .$
Finally,
$C_S = \frac{RM_{in} - A (V_S(0)+(RV_{in}-RV_{out})t)^{\frac{RV_{in}}{(RV_{out}-RV_{in})}}}{RV_{in}} \; ,$
or with the formula for $A$ substituted in
$C_S(t) = \frac{RM_{in} - (RM_{in}-C_S(0)\cdot RV_{in}) (1+(\frac{RV_{in}-RV_{out}}{V_S(0)})t)^{\frac{RV_{in}}{(RV_{out}-RV_{in})}}}{RV_{in}} \; .$
Hope this helps. Sorry for the longwinded derivation with little text and too many formulas. Feel free to ask questions if something is unclear.