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This is the Sylvester equation. A simple explicit solution is possible under certain conditions (no common eigenvalues of the matrices $C^{-1}A$ and $-DB^{-1}$), as explained in the Wikipedia page. There are more complicated general methods, see

1. The explicit solution of the matrix equation $AX−XB=C$ (1995)
2. Continued-fraction solution of matrix equation $AX-XB=C$(1989)
3. Explicit solutions of the matrix equation $AX−XB=C$ (1974)
4. Explicit Solutions of Linear Matrix Equations (1970)