48 Solving the basic linear differential equation

Suppose that $A$ is a constant $(n\times n)$ matrix and that $U(t)$ is a $(n\times k)$ matrix with continuous functions $[0,\mathrm{\infty })\to 𝐂$ as entries. Then for any constant $(n\times k)$ complex matrix $X_0$, the $(n\times k)$ matrix function

$$X(t)=\exp (tA)X_0+{\int }_0^t\exp ((t-s)A)U(s)𝑑s$$

satisfies the matrix differential equation

$$\frac{dX}{dt}=AX+U$$

with initial value

$$X(0)=X_0.$$