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

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Energy conservation property

The model u^{\prime\prime} + \omega^2 u = 0,\quad u(0)=I,\ u^{\prime}(0)=V, has the nice energy conservation property that E(t) = \half(u^{\prime})^2 + \half\omega^2u^2 = \hbox{const}\tp This can be used to check solutions.

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