$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Extension to a variable coefficient; Forward and Backward Euler
$$
\begin{equation}
u'(t) = -a(t)u(t),\quad t\in (0,T],\quad u(0)=I
\tag{32}
\end{equation}
$$
The Forward Euler scheme:
$$
\begin{equation}
\frac{u^{n+1} - u^n}{\Delta t} = -a(t_n)u^n
\end{equation}
$$
The Backward Euler scheme:
$$
\begin{equation}
\frac{u^{n} - u^{n-1}}{\Delta t} = -a(t_n)u^n
\end{equation}
$$
