Point transformations and classification of third-order linear ODE (Q1264836)

The author proves that any third-order linear ODE is local equivalent to one of the following equations: \[ y'''=0, \quad y'''= Ky'+y, \] \[ y'''= \left[ xu^2-2 \left({u' \over u} \right)'+ \left({u' \over u} \right)^2 \right] y'+{1 \over 2}\left[ \left(xu^2-2 \left({u'\over u} \right)'+ \left({u' \over u} \right)^2 \right)'-u^3 \right]y, \] where \(K\in\mathbb{R}\) and \(u\) is a nonvanishing, smooth function of \(x\). Reviewer's remark: Global equivalence of linear ODE was considered by \textit{F. Neuman} [Global properties of linear ordinary differential equations, Kluwer Academic Publishers, Dordrecht (1991; Zbl 0784.34009)]. Here, a criterion of global equivalence of third and higher-order equations is given. A survey of canonical forms of third-order ODEs is presented as well.