Ordinary di
ff
erential equations
Linear di
ff
erential equations and systems
Nonhomogeneous linear equations and systems
Chapters 124: Ordinary Di
ff
erential Equations
Sections 1.1, 1.7, 2.2, 2.6, 2.7, 4.2 & 4.3
Chapters 124: Ordinary Di
ff
erential Equations
Ordinary di
ff
erential equations
Linear di
ff
erential equations and systems
Nonhomogeneous linear equations and systems
Definitions
Existence and uniqueness of solutions
1. Ordinary di
ff
erential equations
An
ordinary di
ff
erential equation
of order
n
is an equation of
the form
d
n
y
dx
n
=
f
x
,
y
,
dy
dx
, . . . ,
d
n

1
y
dx
n

1
.
(1)
A
solution
to this di
ff
erential equation is an
n
times
di
ff
erentiable function
y
(
x
) which satisfies (1).
Example:
Consider the di
ff
erential equation
y

2
y
+
y
= 0
.
What is the order of this equation?
Are
y
1
(
x
) =
e
x
and
y
2
(
x
) =
x e
x
solutions of this di
ff
erential
equation?
Are
y
1
(
x
) and
y
2
(
x
) linearly independent?
Chapters 124: Ordinary Di
ff
erential Equations
Ordinary di
ff
erential equations
Linear di
ff
erential equations and systems
Nonhomogeneous linear equations and systems
Definitions
Existence and uniqueness of solutions
Initial and boundary conditions
An
initial condition
is the prescription of the values of
y
and
of its (
n

1)st derivatives at a point
x
0
,
y
(
x
0
) =
y
0
,
dy
dx
(
x
0
) =
y
1
, . . .
d
n

1
y
dx
n

1
(
x
0
) =
y
n

1
,
(2)
where
y
0
,
y
1
, ...
y
n

1
are given numbers.
Boundary conditions
prescribe the values of linear
combinations of
y
and its derivatives for two di
ff
erent values
of
x
.
In
MATH 254
, you saw various methods to solve ordinary
di
ff
erential equations. Recall that initial or boundary
conditions should be imposed
after
the general solution of a
di
ff
erential equation has been found.
Chapters 124: Ordinary Di
ff
erential Equations
Ordinary di
ff
erential equations
Linear di
ff
erential equations and systems
Nonhomogeneous linear equations and systems
Definitions
Existence and uniqueness of solutions
2. Existence and uniqueness of solutions
Equation (1) may be written as a
firstorder system
dY
dx
=
F
(
x
,
Y
)
(3)
by setting
Y
=
y
,
dy
dx
,
d
2
y
dx
,
· · ·
,
d
n

1
y
dx
n

1
T
.
Existence and uniqueness of solutions:
if
F
in (3) is
continuously di
ff
erentiable in the rectangle
R
=
{
(
x
,
Y
)
,

x

x
0

<
a
,

Y

Y
0

<
b
,
a
,
b
>
0
}
,
then the initial value problem
dY
dx
=
F
(
x
,
Y
)
,
Y
(
x
0
) =
Y
0
,
has a solution in a neighborhood of (
x
0
,
Y
0
). Moreover, this
solution is unique.
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 Spring '08
 STAFF
 Math, Linear Equations, Equations, Elementary algebra, ORDINARY DIFFERENTIAL EQUATIONS, yh

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