Department of Mathematical Sciences
Carnegie Mellon University
21-476 Ordinary Differential Equations Fall 2003
O. Introduction
When modelled mathematically, many real-world phenomena lead to problems in
which one would like to determine a function ufrom an equation involving uand
one or more of its derivatives. Such an equation is called a differential equation. The
unknown function umay be scalar-valued or vector-valued and it may be a function
of a single (scalar) variable or of many variables.
In this course we shall study ordinary differential equations, i.e. differential equa-
tions in which the unknown is a function of a single (real) variable. We shall usually
think of the independent variable as being time (and denote it by t). Of course, in
many applications, the dependent variable may have a different interpretation. We
shall focus our attention primarily on first-order systems of the form
(DE) ˙x(t) = f(t, x(t)),
where f:R×Rn→Rnis a given function and the unknown xtakes values in Rn.
The system (DE) is more general than it may appear at first glance. Indeed, a
large class of higher-order equations and systems can be put in the form (DE) by a
simple change of variables. For example, the second-order scalar equation
(0.1) ¨
θ(t) + α˙
θ(t) + g
lsin θ(t) = 0,
which models the motion of a damped pendulum, can be put in the form (DE) (with
n= 2) by letting x1=θand x2=˙
θ. Similarly the system
(0.2)
u(3)(t) + ¨u(t) + t˙v(t)−u(t) = e3t
¨v(t) = 2t˙v(t) + t3u(t) = v(t) = cos t
of two scalar equations can be put in the form (DE) (with n= 5) by letting x1=
u, x2= ˙u, x3= ¨u, x4=v, x5= ˙v. Here u(3) is the third derivative of u.]
In applications, solutions of differential equations are typically required to satisfy
certain auxiliary conditions, such as initial conditions or boundary conditions. Due
to time limitations, the only type of auxiliary conditions that will be discussed in this
course are initial conditions, i.e. conditions of the form
(IC) x(t0) = x0
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