$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
The \( \theta \)-rule
$$
\begin{equation}
\frac{u^{n+1}-u^n}{\Delta t} = \theta f(u^{n+1},t_{n+1}) +
(1-\theta)f(u^n, t_n)
\tag{38}
\end{equation}
$$
Bringing the unknown \( u^{n+1} \) to the left-hand side and the known terms
on the right-hand side gives
$$
\begin{equation}
u^{n+1} - \Delta t \theta f(u^{n+1},t_{n+1}) =
u^n + \Delta t(1-\theta)f(u^n, t_n)
\end{equation}
$$
This is a nonlinear equation in \( u^{n+1} \) (unless \( f \) is linear in \( u \))!
