In this thesis we present an approach to the approximate solution of a class of nonlinear autonomous systems. We use a linear time-varying model, x + a(t)x = 0 to approximate the nonlinear system x + f(x) = 0, i.e., the solution of the former system is a good approximation to the solution of the latter one. The function a(t) will depend on the nature of f(x) and on the initial condition x(0). If we choose a(t) to be a constant, then this method reduces to the conventional iterative method which uses the linear time-invariant model lc + ax = 0 as an approximation to the nonlinear system. We may choose a(t) in several ways. Here we assume a form with an undetermined parameter for a(t) and then we determine the parameter by matching the trajectories of both systems in the phase plane. This method can also be applied to second order system with some modification. Excellent results are obtained when applied to specific examples. Also it is a promising idea to extend this method to driven nonlinear systems.