  We study the productions of first radial excited states $D_q(2S)$ ($q=u, d,
s$) and $B_q(2S)$ in exclusive semi-leptonic $B_{q'}$ ($q'=u, d s, c$) decays
by the improved Bethe-Salpeter method. These 2S states can be detected through
their strong decays to ground mesons, where the strong decays are calculated by
the low energy approximation as well as the impulse approximation. Some
channels have ratios of order $10^{-4}$: $Br(B^+\to\bar
D^0(2S)\ell^+{\nu_\ell})\times Br(\bar D^0(2S)\to \bar
D^{*}\pi)\approx(4.9\pm4.0)\times10^{-4}$, $Br(B^0\to
D^-(2S)\ell^+{\nu_\ell})\times Br(D^-(2S)\to \bar
D^{*}\pi)\approx(4.4\pm3.4)\times10^{-4}$. These channels could be measured by
the current B-factories. For $D_s(2S)$, we also obtain a relative large ratio:
$Br(B_s^0\to D_s^-(2S)\ell^+{\nu_\ell})\times Br(D_s^-(2S)\to \bar D^{*}\bar
K)\approx (9.9\pm14.9)\times10^{-4}$. Although there are discrepancies of the
full decay width between the theoretical predictions of ${D}^0(2S)$ and
experimental results of $D(2550)^0$, the new detected state $D(2550)^0$ is very
likely the ${D}^0(2S)$ state.
