Linearization of higher order ordinary differential equations

Abstract

Lie’s method for converting a scalar second order ordinary differential equation (ODE) to linear ODE by point transformations was already extended to third and fourth order scalar ODEs by point and contact transformations and to the systems of second order ODEs. The point symmetry group classification of linear nth order scalar and second order systems of m ODEs was provided. Till recently no work on the linearization and classification has been done for higher order systems of ODEs and scalar ODEs linearizable via point, contact and higher order derivative transformations. In this work, we use Meleshko’s algorithm for reducing fourth order autonomous ODEs to second and third order linearizable ODEs and then applying the Ibragimov and Meleshko linearization test for the obtained ODEs. This method can be applied to solve those nonlinear ODEs that are not linearizable by point and contact transformations. Complex-linearization of a class of systems of second order ODEs had been studied with complex symmetry analysis. Linearization of this class had been achieved earlier by complex method, however, linearization conditions and the most general linearizable form of such systems have not been derived yet. It is shown that the general linearizable form of the complex-linearizable systems of two second order ODEs is (at most) quadrat ically semilinear in the first order derivatives of the dependent variables. Linearization conditions for such systems are derived in terms of coefficients of the system and their derivatives. Further, complex methods are employed to obtain the complex-linearizable form of 2−dimensional systems of third order ODEs. This complex-linearizable form leads to a linearizable class of these systems of ODEs. The most general linearizable form and linearization conditions for such class of 2−dimensional systems of third order ODEs are derived with complex-linearization. A canonical form for 2−dimensional linear systems of third order ODEs is obtained by splitting the complex, scalar, third order, linear ODE. This canonical form is used for the symmetry group classification of 2−dimensional linear systems of third order ODEs. Five equivalence classes of such systems with Lie algebras of dimensions 8, 9, 10, 11, and 13 are proved to exist. Contact and higher order derivative symmetries of scalar ODEs are related with the point symmetries of the reduced systems. Two new types of transformations that build up these relations and equivalence classes of scalar third and fourth order ODEs linearizable via these transformations are obtained. Four equivalence classes of these equations are seen to exist.