The prime aim of the present study is to present analytical formulations and solutions for the stability analysis of heterogeneous orthotropic truncated conical shell subjected to external (lateral and hydrostatic) pressures with mixed boundary conditions using the Donnell shell theory. The mixed boundary conditions are as follows: at one end of FGM truncated conical shell is a sleeve that prevents its longitudinal displacement and rotation, and the other end is a freely support. The basic equations of heterogeneous orthotropic truncated conical shells are derived and solved applying the Galerkin's method for the two cases of mixed boundary conditions using new approximation functions. Then the expressions for dimensionless critical external pressures are obtained. The results are compared and validated with the results available in the literature. Finally, a detailed parametric study is conducted to study the effect of heterogeneity, material orthotropy and mixed boundary conditions on the critical external pressures.