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The finite-difference method for solving a nonlinear two-point bound- ary value problems consists of discretization and solving the resulting system of nonlinear equations, usually by the Newton's method. In this paper, we modify Newton's method to solve the system of non- linear equations related to the discretization of a special class of the nonlinear two-point boundary value problems, where y'' is expressed as a multivariate polynomial in y and y'. In the proposed method, the new iteration point is determined as the minimal point of the square of 2-norm of residuals along Newton direction. The minimal point is computed using the solution of a univariate polynomial equation. The numerical examples show that the proposed method performs better than the standard Newton's method with the unit step length. [ABSTRACT FROM AUTHOR]