  We construct the general analytical solution for the $\mathcal{N}$-field
product-exponential potential in an expanding FLRW background. We demonstrate
the relevance of this analytical solution in more general contexts for the
derivation of estimates for the transitional time between an arbitrary initial
state and the slow-roll solutions. In certain cases, these estimates can also
be used to demonstrate the non-linear convergence towards the slow-roll
solutions. In addition, we extend the solution to include stiff matter as well.
