I have non-linear system of ordinary differential equations to solve by first linearising it. I know it can be linearised by expanding right side of equations by Taylor series and keeping only the linear terms. Then I can solve the linear system of differential equations with given initial conditions. Here is the problem. The solution will only be acceptable locally, but not globally. But professor of mine told me to solve the system globally and numerically in this way by finding solution step by step and to compare it with Euler and Runge-Kutta methods for non-linear systems of differential equation. The question is, how can such linearised system be solved numerically step by step, in a way that solution will be "good enough" globally for the given (non-linear) system?