Power Series Solutions of a NODE Systemin the Complex Domain

Abstract

In this Capstone Project, we analyze a second order nonlinear ordinary differential equation (NODE), y^" (x)=f(y^',y) that is impossible to solve analytically. First, using the Taylor Power Series method, we obtain a series expansion of the solution y(x) about x = 0 for x ∈R and find that this series diverges for values of x a little above x = 1. This implies that the equation has a singularity in the complex domain. Therefore, we investigate this NODE by using Laurent expansions about the unknown singularity at x =x_*, which is called movable because its location depends on the initial conditions. By finding the general form of these expansions, we obtain approximate expressions for the singularity closest to x = 0 and thus are able to estimate the radius of convergence for different initial conditions. We also integrate numerically the solutions in the real x, y plane and demonstrate the connection of the global form of the solutions of the problem with the predictions of our laurent series expansions in the complex x- plane.