Considering the Gross-Pitaevskii integral equation we are able to formally obtain an analytical solution for the order parameter $��(x)$ and for the chemical potential $��$ as a function of a unique dimensionless non-linear parameter $��$. We report solutions for different range of values for the repulsive and the attractive non-linear interactions in the condensate. Also, we study a bright soliton-like variational solution for the order parameter for positive and negative values of $��$. Introducing an accumulated error function we have performed a quantitative analysis with other well-established methods as: the perturbation theory, the Thomas-Fermi approximation, and the numerical solution. This study gives a very useful result establishing the universal range of the $��$-values where each solution can be easily implemented. In particular we showed that for $��

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