Invariants of third-order ordinary differential equations $y^{\prime \prime \prime }=f(x,y,y^{\prime },y^{\prime \prime })$ via fiber preserving transformations

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 $y^{\prime \prime \prime }=f(x,y,y^{\prime },y^{\prime \prime })$ 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 $\bar{x}=\varphi (x),\bar{y}=\psi (x,y)$. 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.