MATH2860U:
Chapter 4
1
HIGHERORDER DIFFERENTIAL
EQUATIONS
Linear Differential Equations: Basic Theory (4.1, pg. 118)
Recall:
So far, we’ve only discussed firstorder equations.
Let us now move on to
solving higherorder differential equations, as we have already seen that these arise
frequently in applications.
Definition:
An
n
th
order linear differential equation
is an equation of the form
)
(
)
(
)
(
)
(
)
(
0
1
1
1
1
x
g
y
x
a
dx
dy
x
a
dx
y
d
x
a
dx
y
d
x
a
n
n
n
n
n
n
If we want to solve it subject to initial conditions
1
0
1
1
0
0
0
)
(
,
,
)
(
,
)
(
n
n
y
x
y
y
x
y
y
x
y
then this is an
n
th
order initialvalue problem
.
Before solving such equations, let’s take some time to develop the theory first.
Theorem:
Let
)
(
),
(
,
),
(
),
(
0
1
1
x
a
x
a
x
a
x
a
n
n
and
)
(
x
g
be continuous on an
interval
I
, and let
0
)
(
x
a
n
for every
x
in this interval.
If
0
x
x
is any point in this
interval, then a solution
)
(
x
y
of the initial value problem
)
(
)
(
)
(
)
(
)
(
0
1
1
1
1
x
g
y
x
a
dx
dy
x
a
dx
y
d
x
a
dx
y
d
x
a
n
n
n
n
n
n
subject to
1
0
1
1
0
0
0
)
(
,
,
)
(
,
)
(
n
n
y
x
y
y
x
y
y
x
y
exists and is unique.
Example:
What is the largest interval on which
0
)
5
(
)
3
(
2
y
t
y
y
t
t
subject to
6
)
1
(
,
0
)
1
(
y
y
is guaranteed to have a unique solution?
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Chapter 4
2
Example:
What is the largest interval on which
x
y
y
x
y
x
y
x
tan
2
2
2
3
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 Spring '10
 Kidnan
 Differential Equations, Linear Algebra, Derivative, HigherOrder Diﬀerential Equations

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