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

The Productions and Strong Decays of $D_q(2S)$ and $B_q(2S)$