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CHAPTER 1
Introduction to Diﬀerential Equations
1.1 Basic Terminology
Most of the phenomena studied in the sciences and engineering involve processes that change with
time. For example, it is well known that the rate of decay of a radioactive material at time
t
is
proportional to the amount of material present at time
t
. In mathematical terms this says that
dy
dt
=
ky,
k
a negative constant
(1)
where
y
=
y
(
t
) is the amount of material present at time
t
.
If an object, suspended by a spring, is oscillating up and down, then Newton’s Second Law of
Motion (
F
=
ma
) combined with Hooke’s Law (the restoring force of a spring is proportional to the
displacement of the object) results in the equation
d
2
y
dt
2
+
k
2
y
=0
,k
a positive constant
(2)
where
y
=
y
(
t
) denotes the position of the object at time
t
.
The basic equation governing the diﬀusion of heat in a uniform rod of ±nite length
L
is given
by
∂u
∂t
=
k
2
∂
2
u
∂x
2
(3)
where
u
=
u
(
x, t
) is the temperature of the rod at time
t
at position
x
on the rod.
Each of these equations is an example of what is known as a diﬀerential equation.
DIFFERENTIAL EQUATION
A
diﬀerential equation
is an equation that contains an unknown
function together with one or more of its derivatives.
Here are some additional examples of diﬀerential equations.
Example 1.
(a)
y
±
=
x
2
y

y
y
+1
.
(b)
x
2
d
2
y
dx
2

2
x
dy
dx
+2
y
=4
x
3
.
(c)
∂
2
u
2
+
∂
2
u
∂y
2
= 0
(Laplace’s equation)
(d)
d
3
y
dx
3

4
d
2
y
dx
2
+4
dy
dx
=3
e

x
.
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This note was uploaded on 06/20/2008 for the course M 427K taught by Professor Fonken during the Spring '08 term at University of Texas at Austin.
 Spring '08
 Fonken
 Differential Equations, Equations

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