  The dynamo mechanism is a process of magnetic field self-excitation in a
moving electrically conducting fluid. One of the most interesting applications
of this mechanism related to the astrophysical systems is the case of a random
motion of plasma. For the very first stage of the process, the governing dynamo
equation can be reduced to a system of first order ordinary differential
equations. For this case we suggest a regular method to calculate the growth
rate of magnetic energy. Based on this method we calculate the growth rate for
random flow with finite memory time and anisotropic statistical distribution of
the stretching matrix and compare the results with corresponding ones for
isotropic case and for short-correlated approximation. We find that for
moderate Strouhal numbers and moderate anisotropy the analytical results
reproduce the numerically estimated growth rates reasonably well, while for
larger governing parameters the quantitative difference becomes substantial. In
particular, analytical approximation is applicable for the Strouhal numbers
s<0.6 and we find some numerical models and observational examples for which
this region might be relevant. Rather unexpectedly, we find that the mirror
asymmetry does not contribute to the growth rates obtained, although the mirror
asymmetry effects are known to be crucial for later stages of dynamo action.
