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.