  We compare and discuss the respective efficiency of three methods (with two
variants for each of them), based respectively on Taylor (Maclaurin) series,
Pad\'{e} approximants and conformal mappings, for solving quasi-analytically a
two-point boundary value problem of a nonlinear ordinary differential equation
(ODE). Six configurations of ODE and boundary conditions are successively
considered according to the increasing difficulties that they present. After
having indicated that the Taylor series method almost always requires the
recourse to analytical continuation procedures to be efficient, we use the
complementarity of the two remaining methods (Pad\'{e} and conformal mapping)
to illustrate their respective advantages and limitations. We emphasize the
importance of the existence of solutions with movable singularities for the
efficiency of the methods, particularly for the so-called Pad\'{e}-Hankel
method. (We show that this latter method is equivalent to pushing a movable
pole to infinity.) For each configuration, we determine the singularity
distribution (in the complex plane of the independent variable) of the solution
sought and show how this distribution controls the efficiency of the two
methods. In general the method based on Pad\'{e} approximants is easy to use
and robust but may be awkward in some circumstances whereas the conformal
mapping method is a very fine method which should be used when high accuracy is
required.
