Equivalence transformations of Euler-Bernoulli equation

We give a determination of the equivalence group of Euler-Bernoulli equation and of one of its generalizations, and thus derive some symmetry properties of this equation

Some comparison results on equivalence groups

This paper deals with the comparison of two common types of equivalence groups of differential equations, and this gives rise to a number of results presented in the form of theorems. It is shown in particular that one type can be identified with a subgroup of the other type. Consequences of this comparison related in particular to the determination of invariant functions of the differential equations are also discussed.Comment: 13 page

On point transformations of linear equations of maximal symmetry

An effective method for generating linear equations of maximal symmetry in their much general normal form is obtained. In the said normal form, the coefficients of the equation are differential functions of the coefficient of the term of third highest order. As a result, an explicit expression for the point transformation reducing the equation to its canonical form is obtained, and a simple formula for the expression of the general solution in terms of those of the second-order source equation is recovered. New expressions for the general solution are also obtained, as well as a direct proof of the fact that a linear equation is iterative if and only if it is reducible to the canonical form by a point transformation. New classes of solvable equations parameterized by arbitrary functions are also derived, together with simple algebraic expressions for the corresponding general solution.Comment: 12 pages, Original research pape

Characterization of the class of canonical forms for systems of linear equations

The equivalence group is determined for systems of linear ordinary differential equations in both the standard form and the normal form. It is then shown that the normal form of linear systems reducible by an invertible point transformation to the canonical equation \by^{(n)}=0 consists of copies of the same iterative equation. Other properties of iterative linear systems are also derived, as well as the superposition formula for their general solution.Comment: 14 pages; Original research pape

On generating relative and absolute invariants of linear differential equations

A general expression for a relative invariant of a linear ordinary differential equations is given in terms of the fundamental semi-invariant and an absolute invariant. This result is used to established a number of properties of relative invariants, and it is explicitly shown how to generate fundamental sets of relative and absolute invariants of all orders for the general linear equation. Explicit constructions are made for the linear ODE of order five. The approach used for the explicit determination of invariants is based on an infinitesimal method.Comment: 11 page

Invariants associated with linear ordinary differential equations

We apply a novel method for the equivalence group and its infinitesimal generators to the investigation of invariants of linear ordinary differential equations. First, a comparative study of this method is illustrated by an example. Next, the method is used to obtain the invariants of low order linear ordinary differential equations, and the structure invariance group for an arbitrary order of these equations. Other properties of these equations are also discussed, including the exact number of their invariants.Comment: 12 page