Logistic Solution of Haldane-Radic Equation

Despite the formal solution (1.158) was achieved, it still suffers from a lack in analytical shape since the IT-Lambert poses an implicit functional character. In order to improve such implicit solution one can consider in Eq. (1.157) the same binomial to exponential transformation for initial substrate [SJ as previously performed for the instantaneous free substrate [5](0, see Eqs. (1.151) and (1.153), viz.  

Equation (1.160) may be turned into an analytical expression once the logistic transformation is employed according with the general recipe of Eq. (1.57) to provide the logistic substrate temporal form  

Worth noting that the logistic solution (1.162) finely tunes the extreme conditions for the substrate kinetics, namely  

From now on, with the help of expression (1.162) the temporal course of the kinetics (1.135) may be formulated in an analytical maimer by employing the required temporal derivative of the substrate (Putz, 2011)  

the initial velocity for product formation, i.e., at t = 0, is (Putz, 2011)  

---