We introduce a new class of symmetries, that strictly includes Lie symmetries, for which there exists an algorithm that lets us reduce the order of an ordinary differential equation. Many of the known order-reduction processes, that are not consequence of the existence of Lie symmetries, are a consequence of the invariance of the equation under vector fields of the new class. These vector fields must satisfy a new prolongation formula and there must exist a procedure for determining the vector fields of this class that lead to an equation invariant. We have also found some whose Lie symmetries are trivial, have no obvious order reductions, but can be completely integrated by using the new class of symmetries.
For a given second-order ordinary differential equation (ODE), several relationships among first integrals, integrating factors and λ-symmetries are studied. The knowledge of a λ-symmetry of the equation permits the determination of an integrating factor or a first integral by means of coupled first-order linear systems of partial differential equations. If two nonequivalent λ-symmetries of the equation are known, then an algorithm to find two functionally independent first integrals is provided. These methods include and complete other methods to find integrating factors or first integrals that are based on variational derivatives or in the Prelle–Singer method. These results are applied to several ODEs that appear in the study of relevant equations of mathematical physics.
The class of nonlinear second-order equations that are linearizable by means of generalized Sundman transformations (S-transformations) is identified as the class of equations admitting first integrals that are polynomials of first degree in the first-order derivative. This class is also characterized in terms of the coefficients of the equations and constructive methods to derive the linearizing S-transformations are presented. Only the equations of a well-defined subclass can also be linearized by invertible point transformations. These invertible point transformations can be constructed by using the algorithms for the calculation of linearizing S-transformations. Several examples illustrate that both types of linearization are strictly different.
We investigate the relationship between integrating factors and λ −symmetries for ordinary differential equations of arbitrary order. Some results on the existence of λ −symmetries are used to prove an independent existence theorem for integrating factors. A new method to calculate integrating factors and the associated first integrals is derived from the method to compute λ −symmetries and the associated reduction algorithm. Several examples illustrate how the method works in practice and how the computations that appear in other methods may be simplified.
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