In this paper, by using some block operator matrix techniques, we find explicit solution of the operator equation $TXS^* -SX^*T^*=A$ in the general setting of the adjointable operators between Hilbert $C^*$-modules. Furthermore, we solve the operator equation $TXS^* -SX^*T^*=A$, when $\rm{ran}(T)+\rm{ran}(S)$ is closed.