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arXiv · 1410.0815

Invariants of third-order ordinary differential equations $y'''=f(x,y,y',y'')$ via fiber preserving transformations

Ahmad Y. Al-Dweik·M. T. Mustafa·H. Azad·F. M. Mahomed
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Abstract

Bagderina \cite{Bagderina2008} solved the equivalence problem for scalar third-order ordinary differential equations (ODEs), quadratic in the second-order derivative, via point transformations. However, the question is open for the general class $y'''=f(x,y,y',y'')$ which is not quadratic in the second-order derivative. We utilize Lie's infinitesimal method to study the differential invariants of this general class under pseudo-group of fiber preserving equivalence transformations $\bar{x}=ϕ(x), \bar{y}=ψ(x,y)$. As a result, all third-order differential invariants of this group and the invariant differentiation operators are determined. This leads to simple necessary explicit conditions for a third-order ODE to be equivalent to the respective canonical form under the considered group of transformations. Applications motivated by the literature are presented.

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Ahmad Y. Al-Dweik, M. T. Mustafa, H. Azad, F. M. Mahomed. 2014-10-03. Invariants of third-order ordinary differential equations $y'''=f(x,y,y',y'')$ via fiber preserving transformations. https://arxiv.org/abs/1410.0815

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