$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Systems of ODEs
$$
\begin{align*}
\frac{d}{dt}u_0(t) &= f_0(u_0(t),u_1(t),\ldots,u_N(t),t)\\
\frac{d}{dt}u_1(t) &= f_1(u_0(t),u_1(t),\ldots,u_N(t),t),\\
&\vdots\\
\frac{d}{dt}u_N(t) &= f_N(u_0(t),u_1(t),\ldots,u_N(t),t)
\end{align*}
$$
Introduce vector notation:
- \( u=(u_0(t),u_1(t),\ldots,u_N(t)) \)
- \( (f_0(u,t),f_1(u,t),\ldots,f_N(u,t)) \)
Vector form:
$$ u^{\prime} = f(u,t),\quad u(0)=U_0 $$
Schemes: apply scalar scheme to each component
