The classification of linear ordinary differential equations. I (Q1873493)

This paper is concerned with the classification of \(n\)th-order \((n\geq 3)\) linear ODEs with respect to contact transformations. Using the coincidence of contact symmetries with point symmetries for \(n> 3\) and the invariance of the number of point symmetries under contact transformations for \(n\geq 3\) (see, the author [Russ. J. Math. Phys. 4, 403--407 (1996; Zbl 0913.34033) and Differ. Geom. Appl. 6, 343--350 (1996; Zbl 0879.34009)]), the classification is reduced to the cases of \(n+ 4\), \(n+ 2\) or \(n+ 1\) point symmetries, respectively. The following results are proved: Every linear \(n\)th-order ODE \((n\geq 3)\) with \(n+ 4\) point symmetries is equivalent to \(y^{(n)}=0\). Every contact transformation of two \(n\)th-order ODEs with \(n+ 2\) or \(n+ 1\) point symmetries is a lift of a point transformation. A linear \(n\)th-order ODE with \(n+ 2\) point symmetries may be locally (in a neighborhood of a regular point) transformed to an equation with constant coefficients. In this case, certain sets of numbers (given by quotients of the constant coefficients), as complete sets of invariants, are given.