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