The steady nonequilibrium space charge in concentration polarization at a permselective homogeneous interface

The steady nonequilibrium space charge in concentration polarization at a permselective homogeneous interface [2]. It 

In order to trace this phenomenon in more detail and to investigate some properties of the space charge that forms, consider the following space charge analogue of (4.4.1) 

Here j is again an unknown integration constant (dimensionless cation flux, electric current density) to be determined from the boundary conditions (5.3.8). 

Let us point out first that the system in (5.3.6), (5.3.7) can be integrated exactly in terms of the Painleve transcendents of the second kind. Indeed, by going back to (5.3.1)-(5.3.3) we observe that adding (5.3.1), (5.3.2) and substituting equation yields 

Furthermore, after another integration we obtain, using (5.3.6b), 53 9) 

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