Chapter 9.6, Problem 10P

Problems

Let $X\left(t\right)$ be a fundamental matric for the system $x^{\prime }=A\left(t\right)x\left(t\right),$ where $A\left(t\right)$ is an $n\times n$ matrix function. Show that the solution to the initial-value problem

$x^{\prime }\left(t\right)=A\left(t\right)x\left(t\right)+b\left(t\right),x\left(t_0\right)=x_0$

can be written as

$x\left(t\right)=X\left(t\right)X^{-1}\left(t_0\right)x_0+X\left(t\right){\displaystyle {\int }_{t_0}^t{X}^{-1}\left(s\right)b\left(s\right)ds}$.