ODE Basics
The rest is important details
Stephanos Venakides
August 27, 2009
Motivation
This outline aims at a basic initial contact with prominent concepts and calculations of
the subject of ordinary differential equations (ODE). The intention is that the student will
approach additional material as an elaboration into an already familiar global view.
The
outline also makes a connection between the ODE material taught in MTH107 with the
ODE material of MTH108.
What is an ODE or a system of ODE?
Ordinary differential equations and systems of such equations are often considered in a form
that describes the motion of a particle in a space called
phasespace
. In applications, phase
space is generally not the physical, geometric space. Its coordinates may quantify physical,
chemical, biological, financial, social or other entities. Similarly, the “time” variable
t
may
not stand for time at all. It may represent some other variable on which the phasespace
coordinates depend.
The mathematical analysis of ODE is independent of the context of
particular applications. Nevertheless, the intuition of a particle moving in physical space is
helpful and is pursued here.
An ordinary differential equation (or the system of equations) in phasespace form ex
presses the particle velocity in terms of its position; inserting the coordinates of the position
of the particle in the equation or system produces the value of the particle’s velocity. For ex
ample, in the case of a 2dimensional phasespace, or
phaseplane
, with coordinates (
y
1
, y
2
),
the ODE system is
dy
1
dt
=
V
1
(
y
1
, y
2
)
dy
2
dt
=
V
2
(
y
1
, y
2
)
.
(1)
The functions
V
1
(
y
1
, y
2
) and
V
2
(
y
1
, y
2
), which give the velocity in terms of the position, are
assumed to be given. The intuition is that, as the particle changes position, its velocity is
continuously updated by the ODE and the motion is, thus, specified. A more precise version
of this statement is a mathematical theorem, if sufficient
smoothness
of the velocity field is
assumed. The smoothness condition is satisfied, for example, if the expression for velocity
1
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is continuously differentiable with respect to the coordinates of phasespace. The system of
ODE in an
n
dimensional phasespace is often written in vector form
d
y
dt
=
V
(
y
)
,
(2)
where,
y
=
y
1
.
.
y
n
,
V
=
V
1
.
.
V
n
.
(3)
Each of the components of the velocity vector depends on the position coordinates
y
1
,
· · ·
, y
n
but not on time. The function
V
=
V
(
y
) is a
vector field
, that is “pluggingin” the vector
y
produces another vector,
V
.
A large part of the material of MTH107 consists of the solution of such systems in the
special case in which the velocity depends on the position linearly,
d
y
dt
=
A
y
.
(4)
Here,
A
is a constant square matrix that multiplies the position vector
y
. Solving this system
requires the calculation of the eigenvalues and corresponding eigenvectors of the matrix. In
two dimensions, the system of equations is
dy
1
dt
=
a
11
y
1
+
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 Fall '07
 Trangenstein
 Equations, Ode, Constant of integration, Boundary value problem

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