@Carl Love To be honest, I do not know, but they are methods that we use during our lectures of ODE most of the time. Yes, the first part is solution with using psi=100, so it is linear system. And for the second part, I have to use psi=10*sqrt(dA(z)) and I obtain non-linear system then. This non-linear system I have to deal with I should solve with mentioned shooting method and 4th order RK method. 

@tomleslie First of all, I want to thank you, but probably it is my fault that your solution does not satisfy me. Below, you can find the task for my problem (because there is not problem with final values for cA(t), but for dA(t) I cannot get negative values (it is physical non-sense):

Concentration profiles cA,dAcA,dA during extraction of A-component from solvent C into solvent B in countercurrent colony with axial mixing of continual phase of solvent B are described with diff equations (previous post) which I have to solve.

cA,dAcA,dA represent concentrations of A-component in original and new solvent, kL,uB,uC,E are constants, "psi" is partition coefficient. Then we have following initial and boundary conditions - z=0 m is inlet of feedstock, z=2 m is inlet of extraction agent.

• cA(0)=100
• uB∗dA(2)+E∗dA′(2)=0
• dA′(0)=0

Firstly, I have to solve it for psi=100 (at this moment system of diff equations is linear) and find value for cA(2) in raffinate then. This part I think I have finished. Then I have to calculate using shooting method and then use proper Runge-Kutta approximation of 4th order (using psi=10∗sqrt(dA), for non-physical option dA =<0 I have to use dA/psi=0).