  We interpret the $X_1(2900)$ as an $S$-wave $\bar{D}_1K$ molecular state in
the Bethe-Salpeter equation approach with the ladder and instantaneous
approximations for the kernel. By solving the Bethe-Salpeter equation
numerically with the kernel containing one-particle-exchange diagrams and
introducing three different form factors (monopole, dipole, and exponential
form factors) in the verties, we find the bound state exists. We also study the
decay width of the decay $X_1(2900)$ to $D^-K^+$.
