HIGHERORDER LINEAR
ORDINARY DIFFERENTIAL EQUATIONS I:
Introduction and Homogeneous Equations
David Levermore
Department of Mathematics
University of Maryland
30 August 2009
Because the presentation of this material in class will differ somewhat from that in
the book, I felt that notes that closely follow the class presentation might be appreciated.
1. Introduction
1.1. Normal Form and Solutions
2
1.2. Initial Value Problems
2
1.3. Intervals of Existence
4
2. Homogeneous Equations: General Theory
2.1. Linear Superposition
5
2.2. Wronskians
8
2.3. Fundamental Sets of Solutions and General Solutions
9
2.4. Linear Independence of Solutions
11
2.5. Reduction of Order
15
3. Homogeneous Equations with Constant Coefficients
3.1. Characteristic Polynomials and the Key Identity
18
3.2. Real Roots of Characteristic Polynomials
19
3.3. Complex Roots of Characteristic Polynomials
22
A. Linear Algebraic Systems and Determinants
A.1. Linear Algebraic Systems
27
A.2. Determinants
28
A.3. Existence of Solutions
30
1
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2
1. Introdution
1.1: Normal Form and Solutions.
An
n
th
order linear ordinary differential equation
can be brought into the linear normal form
d
n
y
dt
n
+
a
1
(
t
)
d
n
−
1
y
dt
n
−
1
+
· · ·
+
a
n
−
1
(
t
)
dy
dt
+
a
n
(
t
)
y
=
f
(
t
)
.
(1.1)
Here
a
1
(
t
)
,
· · ·
, a
n
(
t
) are call
coefficients
while
f
(
t
) is called the
forcing
or
driving
. When
f
(
t
) = 0 the equation is said to be
homogeneous
; otherwise it is said to be
nonhomogeneous
.
Definition:
We say that
y
=
Y
(
t
) is a
solution
of (1.1) over an interval (
t
L
, t
R
)
provided that:
•
the function
Y
is
n
times differentiable over (
t
L
, t
R
),
•
the coefficients
a
1
(
t
),
a
2
(
t
),
· · ·
,
a
n
(
t
), and the forcing
f
(
t
) are defined over
(
t
L
, t
R
),
•
the equation
Y
(
n
)
(
t
) +
a
1
(
t
)
Y
(
n
−
1)
(
t
) +
· · ·
+
a
n
−
1
(
t
)
Y
′
(
t
) +
a
n
(
t
)
Y
(
t
) =
f
(
t
)
is satisfied for every
t
in (
t
L
, t
R
).
The first two bullets simply say that every term appearing in the equation is defined over
the interval (
t
L
, t
R
), while the third says the equation is satisfied at each time
t
in (
t
L
, t
R
).
1.2:
InitialValue Problem.
An
initialvalue problem
associated with (1.1) seeks a
solution
y
=
Y
(
t
) of (1.1) that also satisfies the
initial conditions
Y
(
t
I
) =
y
0
,
Y
′
(
t
I
) =
y
1
,
· · ·
Y
(
n
−
1)
(
t
I
) =
y
n
−
1
,
(1.2)
for some
initial time
(or
initial point
)
t
I
and
initial data
(or
initial values
)
y
0
, y
1
,
· · ·
, y
n
−
1
.
You should know the following basic existence and uniqueness theorem about initialvalue
problems, which we state without proof.
Theorem 1.1 (Basic Existence and Uniqueness Theorem):
Let the func
tions
a
1
, a
2
,
· · ·
, a
n
, and
f
all be continuous over an interval (
t
L
, t
R
). Then given
any initial time
t
I
∈
(
t
L
, t
R
) and any initial data
y
0
, y
1
,
· · ·
, y
n
−
1
there exists
a unique solution
y
=
Y
(
t
) of (1.1) that satisfies the initial conditions (1.2).
Moreover, this solution has at least
n
continuous derivatives over (
t
L
, t
R
). If the
functions
a
1
, a
2
,
· · ·
, a
n
, and
f
all have
k
continuous derivatives over (
t
L
, t
R
) then
this solution has at least
k
+
n
continuous derivatives over (
t
L
, t
R
).
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