How can i express concentration in this ODE ?

Question

So as you can see in this photo, i want to express ci (output concentration of mixture) as a function of time. Qi, Cu, K, xv, V - known facts.

I know this is an easy work, but my math has become rusty.

Problem with expressing ci

Answer 1

$$\frac V{Q_i}\frac {dc_i}{dt}+c_i=\frac {C_uK}{Q_i}X_\nu$$ Multiply both side by $\frac {Q_i}V$ $$\frac {dc_i}{dt}+c_i\frac {Q_i}V=\frac {C_uK}{V}X_\nu$$

It's a first order differential equation

I use method of integration factor with $\mu(t)= e^{\frac {Q_it}V}$

Multiply both sides by $\mu $ $$(c_i e^{\frac {Q_it}V})'=\frac {C_uK}{V}X_\nu e^{\frac {Q_it}V}$$ Integrate both side $$c_i e^{\frac {Q_it}V}=\int \frac {C_uK}{V}X_\nu e^{\frac {Q_it}V} dt$$ Put the cionstants outside of the integral

$$c_i e^{\frac {Q_it}V}=\frac {C_uK}{V}X_\nu \int e^{\frac {Q_it}V}dt$$ Evaluate the integral on the right side

$$c_i e^{\frac {Q_it}V}=\frac {C_uK}{V}X_\nu e^{\frac {Q_it}V}\frac V{Q_i}+C$$ Finally $$c_i =\left ( \frac {{C_uK}X_\nu}{Q_i} +Ce^{\frac {-Q_it}V}\right )$$

Where C is a constant of inetgration to be determinated at $t=0$ for example

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Edit1

Note that you can write the differential equation simply this way

$$\frac {dc_i}{dt}+c_i\frac {Q_i}V=m $$

where

$$m=\frac {C_uK}{V}X_\nu$$ Since you only have a constant on the right side of the equation