Solution for nonhomogenous linear matrix ODE

Question

For matrices $X, A, B\in \mathbb{R}^{n\times n}$, $X = X^\top, B=B^\top$, I am looking to find an explicit solution to the following linear first-order matrix ODE where $A$ and $B$ are constant;

$ \dot{X} = AX + XA^\top + B $

I know that the solution for the homogenous part, $\dot{X} = AX + XA^\top $, is

$X(t) = \exp(tA)X(0)\exp(tA^\top)$

But i don't know how to find the solution for the non-homogenous form. Any help is appreciated.

Answer 1

Hint.

Apply Lagrange's variation of constants. Make a particular solution $X_p = e^{At}C(t)(e^{At})'$ and after substitution into the full ODE we have

$$ \dot C(t) = e^{-At}B( e^{-At})' $$

etc.