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

Abstract

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

Symmetry, Singularities, and Integrability in Complex Dynamics II. Rescaling and Time-Translation in Two-Dimensional Systems

“…Without the constraint on the constants of integration the expression in the square root is simply the general solution of a third-order equation of maximal symmetry, the solution of which can be expressed in terms of the three independent quadratic terms obtainable from the solution set of the corresponding second-order equation [13]. We can transform the second-order differential equation (1.4) into a twodimensional system of first-order equations by a transformation which preserves the symmetries in (1.1).…”

Symmetries, integrals and solutions of ordinary differential equations of maximal symmetry

“…2 There is always some ambiguity with the homogeneity symmetry and sl2 . Here we adopt the standard form for sl2 as presented by Mahomed and Leach [17]; see also Moyo and Leach [20]. The ambiguity of the form of sl2 is highlighted by the differing forms found for nth-order ordinary differential equations of different orders [17] and their fundamental first integrals [6].…”

A Note on the Integrability of a Class of Nonlinear Ordinary Differential Equations

“…which is a particular case of a class of equations considered for their interesting integrability properties in [14].…”