A Simplified Method of Solving Linear and Nonlinear Systems

A simplified method for obtaining the response of linear and nonlinear systems, without knowledge of the roots of the system characteristic equation, is described. The solutions are given as time series representing the values of the response at equally spaced instants of time. Initial conditions are introduced easily through the use of the Laplace transform. It is shown that the Laplace transform of a linear system may be approximated by a z-transform, allowing the time series to be obtained by synthetic division. Two examples for linear constant coefficient systems are worked out including the solution of a third-order system and a firstorder differential equation. The results are compared with the exact solutions obtained by analytic means. The same methods are then extended to the solution of timevarying, nonlinear, and time-lag systems. An example of each type is worked out in detail to illustrate the wide applicability of the technique. A discussion of error considerations is included.