We demonstrate an explicit numerical method for accurate solving the eigenvalue problem for some systems of ordinary differential equations, in particular, those describing electron and hole bound states in semiconductor quantum wells with polynomial potential profiles. Holes states are described by the Luttinger Hamiltonian matrix. For solving the eigenvalue problem we use the recurrent sequences procedure that makes possible to derive exact analytical expression for the eigenfunctions,. Hole bound states energies and corresponding wave functions are calculated in a finite parabolic quantum well as functions of the lateral quasimomentum component and parameters of the potential.