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.
