  We study a system of $A$ identical interacting bosons trapped by an external
field by solving ab initio the many-body Schroedinger equation. A complete
solution by using, for example, the traditional hyperspherical harmonics (HH)
basis develops serious problems due to the large degeneracy of HH basis,
symmetrization of the wave function, calculation of the matrix elements, etc.
for large $A$. Instead of the HH basis, here we use the "potential harmonics"
(PH) basis, which is a subset of HH basis. We assume that the contribution to
the orbital and grand orbital [in $3(A-1)$-dimensional space of the reduced
motion] quantum numbers comes only from the interacting pair. This implies
inclusion of two-body correlations only and disregard of all higher-body
correlations. Such an assumption is ideally suited for the Bose-Einstein
condensate (BEC), which is extremely dilute. Unlike the $(3A-4)$ hyperspherical
variables in HH basis, the PH basis involves only three {\it{active}}
variables. It drastically reduces the number of coupled equations and
calculation of the potential matrix becomes tremendously simplified, as it
involves integrals over only three variables for any $A$. One can easily
incorporate realistic atom-atom interactions in a straight forward manner. We
study the ground and excited state properties of the condensate for both
attractive and repulsive interactions for various particle number.
