Linear Systems of First-Order Differential Equations

In this chapter, the authors discuss linear systems of first-order differential equations. They introduce matrix notation and terminology, review some fundamental facts from matrix theory and linear algebra, and discuss some computational techniques. The authors define the concepts of eigenvalues and eigenvectors of a constant matrix, show how to manually compute eigenvalues and eigenvectors, and illustrate how to use computer software to calculate eigenvalues and eigenvectors. They indicate how to write a system of linear first-order differential equations with constant coefficients using matrix-vector notation and state existence and representation theorems regarding the general solution of both homogeneous and nonhomogeneous linear systems. The authors show how to write the general solution in terms of eigenvalues and eigenvectors when the linear system has constant coefficients.