  In this paper, the results of measurements of the total cross sections for
the reactions $^{10,11,12}$Be+$^{28}$Si and $^{14}$B+$^{28}$Si in the beam
energy range $13A$--$47A$ MeV are presented. The experimental cross sections
were obtained by detection of the gamma quanta and neutrons accompanying the
interaction of the isotopes of Be and B with $^{28}$Si. It was found that the
cross sections for $^{11,12}$Be are similar, but significantly exceed those for
$^{10}$Be. A significant increase in the cross sections for $^{12}$Be with
decreasing energy is observed in the entire measured energy range. A
theoretical explanation of the obtained experimental data is given based on the
microscopic model of deformed nuclei and the numerical solution of the
time-dependent Schr\"{o}dinger equation for the outer weakly bound neutrons of
the projectile nuclei. The calculated total reaction cross sections are in good
agreement with the experimental data.
