164 Ordinary Differential Equationsnonhomogeneous linear differential equation and 4yC^2 ) - 3xy = 0 is a second-
order homogeneo us linear differential equation. The second-order differential
equation yC^2 ) + yy(l) - 2y = 0 is not linear b ecause of the term yyCl). The
differential equation (yC^2 l)^3 + (sinx)y = 3ex is not linear b ecause of the t erm
(yC2) )3.
In general, for n;:::: 2 then-th order linear differential equ ation (1) cannot b e
solved expli citly in terms of elementary functions or written as a formula in-
volving quadratures as it can in the case when n = l. Nonetheless, many phys-
ical phenomena such as mechanical systems and electrical circuits can be mod-
elled by n-th order linear differential equations. Consequently, n-th order lin-
ear differential equations are important in the study of physics, engineering,
and other appli ed sciences. Linear differential equations are "first order"
("lowest order") mathematical a pproximations to a wide variety of physical
problems. In section 4.1, we discuss t he basic theory for n-th order linear
differential equations. We state conditions which ensure the existence and
uniqueness of solutions to (1). When (1) is homogeneous, we prove a super-
position theorem which tells us how to combine solut ions to obtain other,
more general, solutions. We define the concept of linear independence for a
set of functions and prove when (1) is homogeneous that there are n linearly
independent solutions and show how to write the general solution in terms
of those linearly indep endent solutions. Finally, we show how to write the
general solution of (1) when it is nonhomogeneous.
In section 4.2, we present a brief history of the search for methods to find
roots of polynomial equations. Next, in section 4.3, we show how to find
t he general solution of an n-th order homogeneous linear differential equation
with constant coefficients by calculating the roots of an n-th degree polynomial
equat ion. Then in section 4.4, we indicate how to find the general solution of
a nonhomogeneous linear differential equation with constant coefficients using
the method of undetermined coefficients. In chapter 5, we present t he Laplace
transform method for solving nonhomogeneous linear differential equations
with constant coefficients. In chapt er 6, we present several a pplications whose
solution ultimately requires the solution of some n-th order linear differential
equation with constant coefficients. Later in chapter 7, which concerns the
solution of systems of n first-order differential equations, we show how to solve
a general n-th order linear different ial equation by rewriting it as a system of
n first-order equ ations.
4.1 Basic Theory
The initial value problem associated with the n-th order linear differential
equation (1) is defined as follows.