Numerical methods to solve nonlinear boundary value ordinary differential equations

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I am dealing with a second order and a 10th order nonlinear differential equation. I want to solve these problems using the Petrov-Galerkin method, but for that I need to linearize the differential equations first. How can I linearize a higher order nonlinear ordinary differential equation to solve with the Petrov-Galerkin method? I have heard about the quasilinearization process developed by Bellman and Kalaba. Can anyone suggest me good resources to learn the quasilinearization process or show me the way to conduct the process?