This work investigates the application of the Newton's method for the numerical solution of a nonlinear boundary value problem formulated through an ordinary differential equation (ODE). Nonlinear ODEs arise in various mathematical modeling contexts, where an exact solution is often unfeasible due to the intrinsic complexity of these equations. Thus, a numerical approach is employed, using Newton's method to solve the system resulting from the discretization of the original problem. The procedure involves the iterative formulation of the method, which enables the approximation of solutions and the evaluation of convergence with respect to the problem parameters. The results demonstrate that Newton's method provides a robust and efficient solution, highlighting its applicability to complex boundary value problems and reinforcing its relevance for the numerical analysis of nonlinear systems. It is concluded that the methodology discussed is suitable for solving a wide range of boundary value problems, ensuring precision and stability in the results.

翻译：本研究探讨了牛顿法在通过常微分方程（ODE）表述的非线性边值问题数值求解中的应用。非线性常微分方程出现在各种数学建模情境中，由于这些方程固有的复杂性，通常难以获得精确解。因此，本文采用数值方法，利用牛顿法求解原始问题离散化后得到的系统。该过程涉及该方法的迭代公式，从而能够近似求解并评估解关于问题参数的收敛性。结果表明，牛顿法提供了一种鲁棒且高效的解决方案，突显了其在复杂边值问题中的适用性，并强化了其在非线性系统数值分析中的重要性。结论表明，所讨论的方法适用于求解广泛的边值问题，确保了结果的精确性和稳定性。