${\displaystyle AM+MB=C}$

Where A, B and C are constant square matrices, and M is the solution that we are trying to find. If A, B, and C are of the same order, and if A and B have no eigenvalues in common, then the solution can be given in terms of matrix exponentials:

${\displaystyle M=-\int _{0}^{\infty }e^{Ax}Ce^{Bx}dx}$