The problem of finding approximate solutions to nonlinear differential equations. (Q1395324)

The author considers a system of ordinary differential equations of the form \[ \dot x=A(x),\qquad x(t_0)=x_0,\tag{1} \] where \(A\) is a known nonlinear operator, \(A:{\mathbb R}^n\longrightarrow{\mathbb R}^n, x\in{\mathbb R}^n, t\in{\mathbb R}^1\), and \(x(t_0)=x_0\) are the initial conditions. He is looking for a solution to (1) in a range of initial values and, next, which is continued to a wider domain by using a pseudo-Taylor factorization of the nonlinear operator \(A(x)\).