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We present results of the first lattice QCD calculations of the weak matrixelements 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 acrossthe entire physical $q^2$ range are then extracted and extrapolated to thecontinuum limit with physical quark masses. Results are derived fromcorrelation functions computed on MILC collaboration gauge configurations withthree different lattice spacings and including 2+1+1 flavours of sea quarks inthe Highly Improved Staggered Quark (HISQ) formalism. HISQ is also used for allof the valence quarks. The uncertainty on the decay widths from our formfactors for $B_c^+ \to D^0 \ell^+ \nu_{\ell}$ is similar in size to that fromthe present value for $V_{ub}$. We obtain the ratio $\Gamma (B_{c}^{+}\rightarrow D^0 \mu^{+} \nu_{\mu}) /\left|\eta_{\mathrm{EW}} V_{ub}\right|^{2}=4.43(63) \times 10^{12} \mathrm{~s}^{-1}$. Combining our formfactors 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 thedifferential 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 possibleeffects from charmonium and $u \overline{u}$ resonances. For example, we findthat 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 $6.31{\tiny}(90)_{B_c \to D_s}(65)_{B_c \to J/\psi} \times 10^{-6}$. We also give resultsfor the branching fraction of $B_c^+ \to D_s^+ \nu \overline{\nu}$.

Form factors for the decay processes $B_c^+ \to D^0 \ell^+ \nu_{\ell}$ and $B_c^+ \to D_s^+ \ell^+ \ell^-$ from lattice QCD