A noniterative method has been developed to solve nonlinear two-point boundary-value problems efficiently and accurately. The original ordinary differential equations are linearized by a parametric differentiation procedure and are then transformed into an initial value problem. Solutions at any value of either an inherent parameter or an artificial parameter can be obtained once first-order, second-order, and remainder terms in a Taylor series expanded with respect to the parameter are computed. To illustrate this new method, mixed-mode convection boundary-layer flows and flames and Troesch's problem are analyzed. Numerical results are presented and discussed. Application of the present method proves most beneficial when (1) solutions of many parametric values are of interest or (1) solutions are highly sensitive to the missing initial conditions.