  We encode the many-body wavefunction of a Bose-Einstein condensate (BEC) in
the $N$-particle sector of an extended catalytic state. This catalytic state is
a coherent state for the condensate mode and an arbitrary state for the modes
orthogonal to the condensate mode. Going to a time-dependent interaction
picture where the state of the condensate mode is displaced to the vacuum, we
can organize the effective Hamiltonian by powers of ${N}^{-1/2}$. Requiring the
terms of order ${N}^{1/2}$ to vanish gives the Gross-Pitaevskii equation. Going
to the next order, $N^0$, we derive equations for the number-conserving
Bogoliubov approximation, first given by Castin and Dum [Phys. Rev. A
$\textbf{57}$, 3008 (1998)]. In contrast to other approaches, ours is well
suited to calculating the state evolution in the Schr\"{o}dinger picture;
moreover, it is straightforward to generalize our method to multi-component
BECs and to higher-order corrections.
