Finding The Lie Symmetries of Some First-Order Odes Via Induced Characteristic (original) (raw)
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An algorithm for solving first order ODEs, by systematically determining symmetries of the form [ξ = F (x), η = P (x) y +Q(x)]-where ξ ∂/∂x+η ∂/∂y is the symmetry generator-is presented. To these 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 always exist but the algorithm will succeed in finding them only partially.
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First-order Ordinary Differential Equations, Symmetries and Linear Transformations1
We present an algorithm for solving first-order ordinary differential equations, by systematically determining symmetries of the form [ξ = F (x), η = P (x) y + Q(x)], where ξ ∂/∂x + η ∂/∂y is the symmetry generator. To these linear symmetries one can associate an ordinary differential equation class which embraces all first-order equations mappable into separable ones through linear transformations {t = f (x), u = p(x) y + q(x)}. This single class includes as members, for instance, 429 of the 552 solvable first-order examples of Kamke's book. Concerning the solution of this class, a restriction on the algorithm being presented exists, only in the case of Riccati equations, for which linear symmetries always exist, but the algorithm will only partially succeed in finding them.
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We present an algorithm for solving first-order ordinary differential equations, by systematically determining symmetries of the form [ξ = F (x), η = P (x) y + Q(x)], where ξ ∂/∂x + η ∂/∂y is the symmetry generator. To these linear symmetries one can associate an ordinary differential equation class which embraces all first-order equations mappable into separable ones through linear transformations {t = f (x), u = p(x) y + q(x)}. This single class includes as members, for instance, 429 of the 552 solvable first-order examples of Kamke's book. Concerning the solution of this class, a restriction on the algorithm being presented exists, only in the case of Riccati equations, for which linear symmetries always exist, but the algorithm will only partially succeed in finding them.
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We investigate the use of invariants of the admitted Lie groups of transformation in finding solutions of the systems of ordinary differential equations (ODEs). Bluman's theorem (1990) of invariant solutions of ODEs is extended for systems of ODEs. Differential invariants of a Lie group are used in reducing order of the given system. Examples are given to illustrate the methods.
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This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Highlights • Two methods to reduce ODEs to generalized Abel or elliptic equations are provided. • New exact solutions for well-known PDEs are obtained. • Computer algebra codes for obtaining the reduced equations are included.