Chapter 8
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 eigen-
vectors of a constant matrix, we show how to manually compute eigenvalues
and eigenvectors, and we illustrate how to use computer software to calcu-
late 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 sys-
tems such as coupled spring-mass systems, pendulum systems, the path of an
electron, and mixture problems.
8.1 Matrices and Vectors
In this section we shall review some facts and computational techniques
from matrix theory and linear algebra. In subsequent sections we will show
how these facts and techniques relate to solving systems of first-order differ-
ential equations.
DEFINITION MatrixA matrix is a rectangular array. We will use a bold-faced capital lettersuch as A , B , C , ... to denote a matrix.
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