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\newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\vex}{{v_{\small\mbox{e}}}} \newcommand{\half}{\frac{1}{2}} \newcommand{\tp}{\thinspace .} \newcommand{\Oof}[1]{\mathcal{O}(#1)}

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Extension to variable coefficients

u'(t) = -a(t)u(t) + b(t)

Forward Euler: \begin{equation} [D_t^+ u = -au + b]^n \tp \end{equation} The truncation error is found from \begin{equation} [D_t^+ \uex + a\uex - b = R]^n \tp \end{equation} Using (9)-(10): \uex'(t_n) - \half\uex''(t_n)\Delta t + \Oof{\Delta t^2} + a(t_n)\uex(t_n) - b(t_n) = R^n \tp Because of the ODE, \uex'(t_n) + a(t_n)\uex(t_n) - b(t_n) =0 , and \begin{equation} R^n = -\half\uex''(t_n)\Delta t + \Oof{\Delta t^2} \tag{32} \tp \end{equation} No problems with variable coefficients!

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