This study provides a step further in the computation of the transition path of a continuous time endogenous growth model discussed by Privileggi (Nonlinear dynamics in economics, finance and social sciences: essays in honour of John Barkley Rosser Jr., Springer, Berlin, Heidelberg, pp. 251---278, 2010)--based on the setting first introduced by Tsur and Zemel (J Econ Dyn Control 31:3459---3477, 2007)--in which knowledge evolves according to the Weitzman (Q J Econ 113:331---360, 1998) recombinant process. A projection method, based on the least squares of the residual function corresponding to the ODE defining the optimal policy of the `detrended' model, allows for the numeric approximation of such policy for a positive Lebesgue measure range of values of the efficiency parameter characterizing the probability function of the recombinant process. Although the projection method's performance rapidly degenerates as one departs from a benchmark value for the efficiency parameter, we are able to numerically compute time-path trajectories which are sufficiently regular to allow for sensitivity analysis under changes in parameters' values.