Let $n\in \mathbb{N}$, let $I\subset \mathbb{R}$ be an interval, let $A:I\to \mathbb{R}^{n\times n}$ and $b:I\to \mathbb{R}^n$ be continuous, and let $\Phi:I\to \mathbb{R}^{n\times n}$ be a fundamental matrix for the homogeneous linear system $\dot{x}=A(t)x$ in the following sense: $\Phi$ is continuous on $I$, differentiable at every interior point of $I$, satisfies $\dot{\Phi}(t)=A(t)\Phi(t)$ at every interior point $t$ of $I$, and $\Phi(t)$ is invertible for every $t\in I$. For every $t_0\in I$ and every $x_0\in \mathbb{R}^n$, define $x:I\to \mathbb{R}^n$ by

\begin{align*} x(t)=\Phi(t)\Phi(t_0)^{-1}x_0+\int_{t_0}^{t}\Phi(t)\Phi(s)^{-1}b(s)\,d\mathcal{L}^1(s), \end{align*}

where the integral is understood with the standard oriented-interval convention. Then $x$ is continuous on $I$, differentiable at every interior point of $I$, satisfies

\begin{align*} \dot{x}(t)=A(t)x(t)+b(t) \end{align*}

at every interior point $t$ of $I$, and satisfies $x(t_0)=x_0$.