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First-order ordinary differential equations, symmetries and linear transformations

  • E. S. CHEB-TERRAB (a1) (a2) and T. KOLOKOLNIKOV (a3)

Abstract

We present an algorithm for solving first-order ordinary differential equations by systematically determining symmetries of the form $[\xi=F(x),\, \eta=P(x)\,y+Q(x)]$, where $\xi\; \pa/\pa x + \eta\; \pa/\pa y$ 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 [12] 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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First-order ordinary differential equations, symmetries and linear transformations

  • E. S. CHEB-TERRAB (a1) (a2) and T. KOLOKOLNIKOV (a3)

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