  We study phase-controlled planar Josephson junctions comprising a
two-dimensional electron gas with strong spin-orbit coupling and d-wave
superconductors, which have an advantage of high critical temperature. We show
that a region between the two superconductors can be tuned into a topological
state by the in-plane Zeeman field, and can host Majorana bound states. The
phase diagram as a function of the Zeeman field, chemical potential, and the
phase difference between superconductors exhibits the appearance of Majorana
bound states for a wide range of parameters. We further investigate the
behavior of the topological gap and its dependence on the type of d-wave
pairing, i.e., d, d+is, or d+id', and note the difficulties that can arise due
to the presence of gapless excitations in pure d-wave superconductors. On the
other hand, the planar Josephson junctions based on superconductors with d+is
and d+id' pairings can potentially lead to realizations of Majorana bound
states. Our proposal can be realized in cuprate superconductors, e.g., in a
twisted bilayer, combined with the layered semiconductor Bi2O2Se.
