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

2014/10/03 by Al-Dweik, Ahmad Y., Mustafa, M. T., Azad, H. +1
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1410.0815

Abstract

Bagderina \citeBagderina2008 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 x=ϕ(x), 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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