Moscow--Petersburg seminar on low-dimensional mathematics
SPb, PDMI, 14.06.2002, 16:00--17:45, room 311   

V.A.Yumaguzhin (Pereslavl-Zalessky)
"On the classification of linear ordinary differential equations".

The aim of this talk is to classify linear ordinary differential
equations (o.d.e.) of order n >= 3 in a neighborhood of a regular
point up to a contact transformation.

It is known that dimension of the algebra of point symmetries of an
n-order linear o.d.e. , n >= 3, is either n+4 or n+2, or n+1.

The following results are proved:
1) any linear o.d.e. with n+4-dimensional algebra of point symmetries
is equivalent to the equation y^{(n)}=0;
2) any linear o.d.e. with n+2-dimensional algebra of point symmetries
is equivalent to some linear o.d.e. with constant coefficients;
constants solving completely the equivalence problem for linear o.d.e.
with constant coefficients are calculated;
3) scalar differential invariants solving completely the equivalence
problem for linear o.d.e. with n+1-dimensional algebra of point
symmetries are calculated.