$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
A Backward Euler scheme for the vector ODE \( u^{\prime}=f(u,t) \)
$$
\begin{align*}
\frac{u_0^n- u_0^{n-1}}{\Delta t} &= f_0(u^n,t_n)\\
\frac{u_1^n- u_1^{n-1}}{\Delta t} &= f_1(u^n,t_n)\\
&\vdots\\
\frac{u_N^n- u_N^{n-1}}{\Delta t} &= f_N(u^n,t_n)
\end{align*}
$$
This can be written more compactly in vector form as
$$ \frac{u^n- u^{n-1}}{\Delta t} = f(u^n,t_n)$$
This is a system of nonlinear algebraic equations,
$$ u^n - \Delta t\,f(u^n,t_n) - u^{n-1}=0,$$
or written out
$$
\begin{align*}
u_0^n - \Delta t\, f_0(u^n,t_n) - u_0^{n-1} &= 0,\\
&\vdots\\
u_N^n - \Delta t\, f_N(u^n,t_n) - u_N^{n-1} &= 0\tp
\end{align*}
$$
