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We study the equivalence problem of classifying second order ordinarydifferential equations $y_{xx}=J(x,y,y_{x})$ modulo fibre-preserving pointtransformations $x\longmapsto \varphi(x)$, $y\longmapsto \psi(x,y)$ by usingMoser's method of normal forms. We first compute a basis of the Lie algebra${\frak{g}}_{{\{y_{xx}=0\}}}$ of fibre-preserving symmetries of $y_{xx}=0$. Inthe formal theory of Moser's method, this Lie algebra is used to give anexplicit description of the set of normal forms $\mathcal{N}$, and we show thatthe set is an ideal in the space of formal power series. We then show theexistence of the normal forms by studying flows of suitable vector fields withappropriate corrections by the Cauchy-Kovalevskaya theorem. As an application,we show how normal forms can be used to prove that the identical vanishing ofHsu-Kamran primary invariants directly imply that the second order differentialequation is fibre-preserving point equivalent to $y_{xx}=0$.

Normal Forms of second order Ordinary Differential Equations $y_{xx}=J(x,y,y_{x})$ under Fibre-Preserving Maps