$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Extension to a source term
$$
\begin{equation}
u'(t) = -a(t)u(t) + b(t),\quad t\in (0,T],\quad u(0)=I
\tag{33}
\end{equation}
$$
$$
\begin{align*}
\lbrack D^+_t u &= -au + b\rbrack^n,\\
\lbrack D^-_t u &= -au + b\rbrack^n,\\
\lbrack D_t u &= -a\overline{u}^t + b\rbrack^{n+\half},\\
\lbrack D_t u &= \overline{-au+b}^t\rbrack^{n+\half}
\end{align*}
$$
