Symmetries, integrals and solutions of ordinary differential equations of maximal symmetry (original) (raw)

Exceptional Properties of Second and Third Order Ordinary Differential Equations of Maximal Symmetry

Journal of Mathematical Analysis and Applications, 2000

The Riccati transformation is used in the reduction of order of second and third Ž . order ordinary differential equations of maximal symmetry. The sl 2, R subalgebra is preserved under this transformation. The Riccati transformation is itself associated with the symmetry that is annihilated in the reduction of order. The solution symmetries and the intrinsically contact symmetries become nonlocal symmetries under the Riccati transformation. We investigate the fate and origins of the contact symmetries arising from the Riccati transformation. The exceptional properties of the second and third order equations of maximal symmetry are indicated. In the context of generalised symmetries we express the solution symmetries, Ž . contact symmetries, and the sl 2, R elements in terms of a Jacobian. We show that a basis for the solution set of equations of maximal symmetry is given in terms of the solution set of a second order ordinary differential equation. ᮊ

Complete Symmetry Groups of Ordinary Differential Equations and Their Integrals: Some Basic Considerations

Journal of Mathematical Analysis and Applications, 2001

The concept of the complete symmetry group of a differential equation introduced by J. Krause (1994, J. Math. Phys. 35, 5734-5748) is extended to integrals of such equations. This paper is devoted to some aspects characterising complete symmetry groups. The algebras of the symmetries of both differential equations and integrals are studied in the context of equations for which the elements are represented by point or contact symmetries so that there is no ambiguity about the group. Both algebras and groups are found to be nonunique. 

Symmetries of First Integrals and Their Associated Differential Equations

Journal of Mathematical Analysis and Applications, 1999

The relationship between the reduction of order through point symmetries and integration is explored with particular emphasis on the loss and gain of point Ž. contact for third order symmetries to and from nonlocal symmetries. It is seen that reduction of order can even lead to the loss of all point symmetries at the third order level and their replacement at the second order level from nonlocal symmetries. It is evident that nonlocal symmetries should be given more attention in applications.

Symmetry Classification of First Integrals for Scalar Linearizable Second-Order ODEs

2012

Symmetries of the fundamental first integrals for scalar second-order ordinary differential equations ODEs which are linear or linearizable by point transformations have already been obtained. Firstly we show how one can determine the relationship between the symmetries and the first integrals of linear or linearizable scalar ODEs of order two. Secondly, a complete classification of point symmetries of first integrals of such linear ODEs is studied. As a consequence, we provide a counting theorem for the point symmetries of first integrals of scalar linearizable second-order ODEs. We show that there exists the 0-, 1-, 2-, or 3-point symmetry cases. It is shown that the maximal algebra case is unique.

Symmetries of linear ordinary differential equations

Journal of Physics A: Mathematical and General, 1997

We discuss the Lie symmetry approach to homogeneous, linear, ordinary di erential equations in an attempt to connect it with the algebraic theory of such equations. In particular we pay attention to the elds of functions over which the symmetry vector elds are de ned and, by de ning a noncharacteristic Lie subalgebra of the symmetry algebra, are able to establish a general description of all continuous symmetries. We use this description to rederive a classical result on di erential extensions for second order equations.

Symmetry and integrability by quadratures of ordinary differential equations

Physics Letters A, 1988

In this paper, the connection between point symmetries and the integrability by quadratures of second-order ordinary differential equations is discussed. An example is given of a family of second-order ordinary differential equations integrable by quadratures whose point symmetry group is, nevertheless, trivial. This refutes the widespread belief that the existence of nontrivial point symmetries is a necessary condition for the integrability by quadratures ofordinary differential equations. The significance ofdynamical (versus point) symmetries in this field is illustrated with a few recent results.

Discrete Symmetry Transformations of Third Order Ordinary Differential Equations and Applications

2020

Third order ordinary differential equations have already been classified by the Lie algebra they admit. Invariant equations corresponding to these Lie algebras are also available in the literature [17]. In this paper, list of all discrete symmetries corresponding to these invariant ordinary differential equations, are obtained. Some particular examples are given to show the significance of the work.

Ordinary Differential Equation: Symmetries and Last Multiplier

Clifford Algebras and their Applications in Mathematical Physics, 2000

This paper shows how to get a last multiplier for a differential n -form equivalent to an ordinary differential equation (ODE) in (n + 1) -dimensions. The last multiplier makes it possible to find a closed differential n -form given by the ODE. Our construction is based on a set of n symmetry vector fields admitted by an ODE. Our result is a generalization of Lie's theorem on integrating factor for an ODE in two dimensions which admits a one symmetry vector field.