Algebraic Properties of First Integrals for Scalar Linear Third-Order ODEs of Maximal Symmetry

Abstract EN

By use of the Lie symmetry group methods we analyze the relationship between the first integrals of the simplest linear third-order ordinary differential equations (ODEs) and their point symmetries.

It is well known that there are three classes of linear third-order ODEs for maximal cases of point symmetries which are 4, 5, and 7.

The simplest scalar linear third-order equation has seven-point symmetries.

We obtain the classifying relation between the symmetry and the first integral for the simplest equation.

It is shown that the maximal Lie algebra of a first integral for the simplest equation y′′′=0 is unique and four-dimensional.

Moreover, we show that the Lie algebra of the simplest linear third-order equation is generated by the symmetries of the two basic integrals.

We also obtain counting theorems of the symmetry properties of the first integrals for such linear third-order ODEs.

Furthermore, we provide insights into the manner in which one can generate the full Lie algebra of higher-order ODEs of maximal symmetry from two of their basic integrals.