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Bagderina \cite{Bagderina2008} solved the equivalence problem for scalarthird-order ordinary differential equations (ODEs), quadratic in thesecond-order derivative, via point transformations. However, the question isopen for the general class $y'''=f(x,y,y',y'')$ which is not quadratic in thesecond-order derivative. We utilize Lie's infinitesimal method to study thedifferential invariants of this general class under pseudo-group of fiberpreserving equivalence transformations $\bar{x}=\phi(x), \bar{y}=\psi(x,y)$. Asa result, all third-order differential invariants of this group and theinvariant differentiation operators are determined. This leads to simplenecessary explicit conditions for a third-order ODE to be equivalent to therespective canonical form under the considered group of transformations.Applications motivated by the literature are presented.

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