A spatial analysis of hydrodynamic instabilities of a non-viscous shear flow was carried out using the Rayleigh stability equation through spectral methods with Chebyshev polynomials. The nonlinear eigenvalue problem that results from solving the differential Rayleigh equation with spectral methods and their respective boundary conditions, was solved numerically providing information on flow instabilities. Finally, A Python code was developed to find the dispersion relations that allowed determining the maximum spatial growth rate of the perturbations.