  A mixture of two kinds of identical bosons held in a harmonic potential and
interacting by harmonic particle-particle interactions is discussed. This is an
exactly-solvable model of a mixture of two trapped Bose-Einstein condensates
which allows us to examine analytically various properties. Generalizing the
treatment in [Cohen and Lee, J. Math. Phys. {\bf 26}, 3105 (1985)], closed form
expressions for the ground-state energy, wave-function, and lowest-order
densities are obtained and analyzed for attractive and repulsive intra-species
and inter-species particle-particle interactions. A particular mean-field
solution of the corresponding Gross-Pitaevskii theory is also found
analytically. This allows us to compare properties of the mixture at the exact,
many-body and mean-field levels, both for finite systems and at the limit of an
infinite number of particles. We hereby prove that the exact ground-state
energy and lowest-order intra-species and inter-species densities converge at
the infinite-particle limit (when the products of the number of particles times
the intra-species and inter-species interaction strengths are held fixed) to
the results of the Gross-Pitaevskii theory for the mixture. Finally and on the
other end, the separability of the mixture's center-of-mass coordinate is used
to show that the Gross-Pitaevskii theory for mixtures is unable to describe the
variance of many-particle operators in the mixture, even in the
infinite-particle limit. Our analytical results show that many-body
correlations exist in a mixture of Bose-Einstein condensates made of any number
of particles. Implications are briefly discussed.
