A Procedure for Determining the Exact Solution to a 22 First Order Homogenous Nonautonomous System

Although numerous attempts have been made by various researchers to find a method to solve the general case, a successful technique remains elusive. For example, a paper by Caviglia and Morro [2] disproves an effort to do so using quadratures. The authors express the notion that such an achievement would be remarkable, but the ability to solve a first order homogeneous nonautonomous system using quadratures is a claim that still remains to be proven. Nonautonomous systems, particularly periodic systems, are of great interest to mathematicians as well researchers from an assortment of scientific and engineering fields. The application of these systems is extensive. For example, electronics circuits are driven by clock signals with a specific period [4]. The dynamics of these circuits can be investigated with an appropriate nonautonomous system as a model. The oscillator is a recurring feature in numerous physical systems. With the use of Floquet and Lyapunov theories the stability of these linear periodic systems can be analyzed. Floquet theory, specifically, governs the structure of the solutions to periodic systems. In this paper the author develops a technique for solving certain 2×2 first order homogeneous nonautonomous systems. While most mathematicians prefer to convert nonlinear differential equations into linear equations when seeking a solution, the method presented as the primary result solves a nonlinear equation in order to construct the exact solution of a linear system. The author’s findings show that the exact solution to a 2×2 nonautonomous system is directly related to the solution of a Riccati differential equation that is constructed from the coefficients in the matrix A(t) of a system x′ = A(t)x, and the general structure of the exact solution to a nonautonomous system is similar to the general form of the exact solution for an autonomous system. However, without a method of solving the general Riccati differential equation, a procedure for determining the exact solution to a general 2×2 nonautonomous system remains elusive. Nevertheless, the main result does develop an approach for finding solutions to many complicated special cases.