First order linear nonautonomous homogenous systems of differential equations are
characterized by a matrix differential equation where the matrix is a function of the
independent variable. These nonautonomous systems are used extensively in the study of
Floquet and Lyapunov theories, and the applications of such systems reaches into fields
such as physics, biology, and engineering. The following paper develops a technique for
finding the closed form solution to a 2×2 nonautonomous system. The paper shows that
the solution to such a system is directly related to the solution of a Riccati differential
equation constructed from the coefficients of the system’s matrix. The primary findings
also demonstrate that the system can be solved exactly if a solution to the corresponding
Riccati equation can be determined.