  We present results of the first lattice QCD calculations of the weak matrix
elements for the decays $B_c^+ \to D^0 \ell^+ \nu_{\ell}$, $B_c^+ \to D_s^+
\ell^+ \ell^-$ and $B_c^+ \to D_s^+ \nu \overline{\nu}$. Form factors across
the entire physical $q^2$ range are then extracted and extrapolated to the
continuum limit with physical quark masses. Results are derived from
correlation functions computed on MILC Collaboration gauge configurations with
three different lattice spacings and including 2+1+1 flavours of sea quarks in
the Highly Improved Staggered Quark (HISQ) formalism. HISQ is also used for all
of the valence quarks. The uncertainty on the decay widths from our form
factors is similar in size to that from the present value for $V_{ub}$. We
obtain the ratio $\Gamma (B_{c}^{+} \rightarrow D^0 \mu^{+} \nu_{\mu})
/\left|\eta_{\mathrm{EW}} V_{u b}\right|^{2}=4.43(63) \times 10^{12}
\mathrm{~s}^{-1}$. Combining our form factors with those found previously by
HPQCD for $B_{c}^{+} \rightarrow J / \psi \mu^{+} \nu_{\mu}$, we find
$\left|V_{cb}/V_{ub} \right|^2 \Gamma( B_c^+ \to D^0 \mu^+ \nu_\mu
)/\Gamma(B_{c}^{+} \rightarrow J / \psi \mu^{+} \nu_{\mu}) = 0.257(36)_{B_c \to
D}(18)_{B_c \to J/\psi}$. We calculate the differential decay widths of $B_c^+
\to D_s^+ \ell^+ \ell^-$ across the full $q^2$ range, and give integrated
results in $q^2$ bins that avoid possible effects from charmonium and $u
\overline{u}$ resonances. For example, we find that the ratio of differential
branching fractions integrated over the range $q^2 = 1 \; \mathrm{GeV}^2 - 6 \;
\mathrm{GeV}^2$ for $B_c^+ \to D_s^+ \mu^+ \mu^-$ and $B_{c}^{+} \rightarrow J
/ \psi \mu^{+} \nu_{\mu}$ is $5.23{\tiny }(73)_{B_c \to D_s}(54)_{B_c \to
J/\psi} \times 10^{-6}$. We also give results for the branching fraction of
$B_c^+ \to D_s^+ \nu \overline{\nu}$. Prospects for reducing our errors in the
future are discussed.
