Elementary Differential Equations with
Boundary Value Problems, 6th ed.
Section
1.1
Differential
Equations
and
Mathematical
Models
C.
Henry
Edwards
David
E.
Penney
Objectives
Introduction
Differential Equations
Example
Goals of the Study of Differential Equations
Example
Objectives
Differential Equations and Mathematical Models
Introduction
Examples
Infinite Family of Solutions
Example
Initial Conditions
Mathematical Models
A ThreeStep Cycle
Mathematical Models
Objectives (cont’d)
Examples and Terminology
Examples
Existence and Uniqueness
Terminology
Examples
Ordinary and Partial Differential Equations
Initial Value Problems
Example
Finding Solutions
Introduction
Differential Equations
Changing Quantities
•
The laws of the universe are written in the language of
mathematics.
•
Algebra is sufficient to solve many static problems, but the
most interesting natural phenomena involve change and are
described by equations that relate changing quantities.
Differential Equations
Derivative as Rate of Change
•
Because the derivative
dx
/
dt
=
f
0
(
t
)
of the function
f
is the
rate at which the quantity
x
=
f
(
t
)
is changing with respect to
the independent variable
t
. . .
•
. . . it is natural that equations involving derivatives are
frequently used to describe the changing universe.
•
An equation relating an unknown function and one or more of
its derivatives is called a
differential equation
.
Example
Example 1
•
The equation
dx
dt
=
x
2
+
t
2
involves the unknown function
x
(
t
)
and its first derivative
x
0
(
t
)
.
•
The equation
d
2
y
dx
2
+
3
dy
dx
+
7
y
=
0
involves the unknown function
y
(
x
)
and its first two
derivatives.
Goals of the Study of Differential Equations
Three Goals
•
The study of differential equations has three principal goals:
1.
To discover the differential equation that describes a
specified physical situation.
2.
To find—either exactly or approximately—the
appropriate solution of that equation.
3.
To interpret the solution that is found.
Goals of the Study of Differential Equations
Unknowns
•
In algebra, we typically seek the unknown
numbers
that satisfy
an equation such as
x
3
+
7
x
2

11
x
+
41
=
0
.
•
By contrast, in solving a differential equation, we are
challenged to find the unknown
functions
y
=
y
(
x
)
for which
an identity such as
y
0
(
x
) =
2
xy
(
x
)
—that is, the differential
equation
dy
dx
=
2
xy
—holds on some interval of real numbers.
•
Ordinarily, we will want to find
all
solutions of the differential
equation, if possible.
Example
Example 2
•
If
C
is a constant and
y
(
x
) =
Ce
x
2
,
then
dy
dx
=
C
2
xe
x
2
= (
2
x
)
Ce
x
2
=
2
xy
.
•
Thus every function
y
(
x
)
of the above form
satisfies
—and
thus is a solution of—the differential equation
dy
dx
=
2
xy
for all
x
.
Example
Example 2 (cont’d)
•
In particular, the equation
y
(
x
) =
Ce
x
2
defines an
infinite
family of different solutions of this
differential equation, one for each choice of the arbitrary
constant
C
.