Does a linear differential equation of a matrix variable have a closed form solution?

Question

Let $X(t), A, B, C$ be matrices and $A, B, C$ are constant matrices. Does the following linear differential equation have a closed form solution? $$ \frac{d X(t)}{dt} = A + BXC. $$ Thank you!

Answer 1

This expands user254433's answer. First, note that $$ \frac{\mathrm{d}}{\mathrm{d}t}\left[\sum_{k\geq0}\frac{t^{k}}{k!}B^{k}MC^{k}\right]=B\left(\sum_{k\geq1}\frac{t^{k-1}}{\left(k-1\right)!}B^{k-1}MC^{k-1}\right)C=B\left(\sum_{k\geq0}\frac{t^{k}}{k!}B^{k}MC^{k}\right)C. $$ Now, suppose $B$ and $C$ are nonsingular. Motivated by the above, take $$ X(t)=\sum_{k\geq0}\frac{t^{k}}{k!}B^{k}\left(X(0)+B^{-1}AC^{-1}\right)C^{k}-B^{-1}AC^{-1} $$ as a solution. You can verify that the above is a solution directly: $$ X^{\prime}(t)=B\left[\sum_{k\geq0}\frac{t^{k}}{k!}B^{k}\left(X(0)+B^{-1}AC^{-1}\right)C^{k}\right]C=A+BX(t)C. $$