  The general conditions under which the quadratic, uniform and monotonic
convergence in the quasilinearization method of solving nonlinear ordinary
differential equations could be proved are formulated and elaborated. The
generalization of the proof to partial differential equations is straight
forward. The method, whose mathematical basis in physics was discussed recently
by one of the present authors (VBM), approximates the solution of a nonlinear
differential equation by treating the nonlinear terms as a perturbation about
the linear ones, and unlike perturbation theories is not based on the existence
of some kind of a small parameter.
  It is shown that the quasilinearization method gives excellent results when
applied to different nonlinear ordinary differential equations in physics, such
as the Blasius, Duffing, Lane-Emden and Thomas-Fermi equations. The first few
quasilinear iterations already provide extremely accurate and numerically
stable answers.
