  The master integrals for the mixed QCD-QED two-loop virtual corrections to
the charged-current Drell-Yan process $q\bar{q}^{\prime} \rightarrow \ell \nu$
are computed analytically by using the differential equation method. A suitable
choice of master integrals makes it successful to cast the differential
equation system into the canonical form. We keep the dependence on charged
lepton mass in the building of differential equations and then expand the
system against the ratio of small charged lepton mass to large $W$-boson mass.
In such a way the final results will contain large logarithms of the form
$\log(m_{\ell}^2/m_W^2)$. Finally, all the canonical master integrals are given
as Taylor series around $d = 4$ spacetime dimensions up to order four, with
coefficients expressed in terms of Goncharov polylogarithms up to weight four.
