Extension to variable coefficients

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!