In this chapter we return to the analysis of systems of two linear autonomous differential equations that we began in Chapter 6. For these autonomous systems, in which the coefficients in the differential equations are all constants, we obtain explicit solutions for all the possible cases and use these solutions to analyze their behavior in the phase plane. In order to characterize this behavior, we introduce the terms node, center, focus, and saddle point to indicate the behavior of solutions near equilibrium points.
We use our graphical analysis to discover how nullclines, which are isoclines for horizontal and vertical tangents in the phase plane, can aid us in determining the stability of equilibrium points. These ideas are important for Chapter 10, where we investigate nonlinear autonomous systems.
We give an alternative method for finding the explicit solutions, one that involves simple properties of 2 by 2 matrices. This alternative method introduces the notions of eigenvalues, eigenvectors, and a fundamental matrix.
Use of these differential equations is illustrated by models of solute movement between two containers, behavior of populations of two countries with mutual emigration, and compartmental models.