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of diverse disciplines which can be written as initial value problems and then
solved using numerical integration software. At the end of the chapter, we
present additional a pplications.
In Chapte r 4 we discuss the basic theory for n-th order linear differential
equations. We present a history of the attempts of mathematicians to find
roots of polynomials. Then we illustrate how to use computer software to
approximate the roots of polynomials numerically. Next, we show how to find
the general solution of an n-th order homogeneous linear differential equation
with constant coefficients by finding the roots of a n n-th degree p olynomial.
Finally, we indicate how to find the general solution of a nonhomogeneous
linear differential equation with constant coefficients using the method of un-
determined coefficients.
In Chapter 5 we define the La place transform and examine its properties.
Next, we show how to solve homogeneous and nonhomogeneo us linear dif-
ferential equations with constant coefficients and their corresponding initial
value problems using the Laplace transform method. Then we define the con-
volution of two functions and prove the convolution theorem. Finally, we show
how to solve nonhomogeneous linear differential equations with constant coef-
ficients in which the nonhomogeneity is a d iscontinuous function, a time-delay
function, or an impulse function.
In Chapter 6 we examine several linear differential equations with constant
coefficients which arise in t he study of various physical and electrical systems.
In Chapte r 7 we define a system of first-order differential equations. We
state a fundamental existence and uniqueness theorem and a continuation
theorem for the system initial value problem. Then , we show how to a pply
these theorems to several initial value problems. Next, we show how to rewrite
an n-th order differential equation as an equivalent system of n first-order
equations.
In Chapte r 8 we discuss linear systems of first-order differential equations.
First, we introduce matrix notation and terminology, we review fundamental
facts from matrix theory and linear algebra, and we discuss some compu-
tational techniques. Next, 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. We show h ow to write a system of linear
first-order differential equations with constant coefficients using matrix-vector
notation, we state existence and representation theorems regarding th e gen-
eral solution of both homogeneous and nonhomogeneous linear systems, and
we show how to write the general solution in terms of eigenvalues and eigen-
vectors.
In Chapte r 9 we examine a few linear systems with constant coefficients
which arise in various physical systems such as coupled spring-mass systems,
pendulum syst ems, the path of an electron, and mixture problems.