First order ODEs, Symmetries and Linear Transformations

Details First order ODEs, Symmetries and Linear Transformations First order ODEs, Symmetries and Linear Transformations

2000-07-15

arxiv.org/abs/math-ph/0007023v1

Physics

Mathematical Physics

13 pages. Submitted to European Journal of Applied Mathematics, July 2000. Related Maple programs are available at http://li

Scientific paper

An algorithm for solving first order ODEs, by systematically determining symmetries of the form [ xi = F(x), eta = P(x) y + Q(x) ], where xi d/dx + eta d/dy is the symmetry generator - is presented. To these {\it linear} symmetries one can associate an ODE class which embraces all first order ODEs mappable into separable through linear transformations {t = f(x), u = p(x) y + q(x)}. This single ODE class includes as members, for instance, 78% of the 552 solvable first order examples of Kamke's book. Concerning the solving of this class, a restriction on the algorithm being presented exists only in the case of Riccati type ODEs, for which linear symmetries {\it always} exist but the algorithm will succeed in finding them only partially.

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