The non-symmetrized hyperspherical harmonics method for a three-body system, composed by two particles having equal masses, but different from the mass of the third particle, is reviewed and applied to the $^3$H, $^3$He nuclei and $^3_{Lambda}$H hyper-nucleus, seen respectively as $nnp$, $ppn$ and $NNLambda$ three-body systems. The convergence of the method is first tested in order to estimate its accuracy. Then, the difference of binding energy between $^3$H and $^3$He due to the difference of the proton and the neutron masses is studied using several central spin-independent and spin-dependent potentials. Finally, the $^3_{Lambda}$H hypernucleus binding energy is calculated using different $NN$ and $Lambda N$ potential models. The results have been compared with those present in the literature, finding a very nice agreement.
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