- Linear Systems of First-Order Differential Equations

In this chapter we discuss linear systems of first-order differential equations. In the first section, Matrices and Vectors, we introduce matrix notation and terminology, we review some fundamental facts from matrix theory and linear algebra, and we discuss some computational techniques. In the second section, Eigenvalues and Eigenvectors, we define the concepts of eigenvalues and eigenvectors of a constant matrix, we show how to manually compute eigenvalues and eigenvectors, and we illustrate how to use computer software to calculate eigenvalues and eigenvectors. In the last section, Linear Systems with Constant Coefficients, we indicate how to write a system of linear first-order differential equations with constant coefficients using matrix-vector notation, we state existence and representation theorems regarding the general solution of both homogeneous and nonhomogeneous linear systems, and we show how to write the general solution in terms of eigenvalues and eigenvectors when the linear system has constant coefficients. In chapter 9, we examine a few linear systems with constant coefficients which arise in various physical systems such as coupled spring-mass systems, pendulum systems, the path of an electron, and mixture problems.