Mixing Question - Linear ODE

[hadroneater]

Homework Statement

A stirred tank reactor that initially contains a volume V(0) = V_0 of water. Suppose that a stirred solution of salt at concentration S is pumped in at a rate of F_in = F litres/hr and the well-stirred mixture is pumped out at a slight faster rate of F_out = (F + f) litres/hr where f > 0. Let C(t) denote the concentration of salt inside the tank. Find C(t).

Homework Equations

The Attempt at a Solution

V(t) = V_0 + F_in*t - F_out*t = V_0 + F*t - F*t + f*t = V_0 + f*t

(C*V)' = C'*V + C*V' = C'(V_0 - f*t) - Cf = SF - C(F + f)
C'(V_0 - f*t) = S*F - C*F
C' + C*F/(V_0 - f*t) = S*F/(V_0 - f*t)

Do I use integration factor to solve for C? I tried it and it was really complicated but the final answer is unusually simple. So I don't think it is right.

 

[Roni1985]

Why don't you try to find C' and not (C*V)' ?

Try, C'=(salt enters)- (salt leaves).

And, Vo is pure water, right ?
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