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Higher Order Linear Equations with Constant Coefficients
The solutions of linear differential equations with constant coefficients of
the third order or higher can be found in similar ways as the solutions of
second order linear equations. For an
n
th order homogeneous linear
equation with constant coefficients:
a
n
y
(
n
)
+
a
n
±1
y
(
n
±1)
+ … +
a
2
y
″ +
a
1
y
′ +
a
0
y
= 0,
a
n
≠ 0.
It has a general solution of the form
y
=
C
1
y
1
+
C
2
y
2
+ … +
C
n
±1
y
n
±1
+
C
n
y
n
where
y
1
,
y
2
, … ,
y
n
±1
,
y
n
are any
n
linearly independent solutions of the
equation.
(Thus, they form a set of fundamental solutions of the differential
equation.)
The linear independence of those solutions can be determined by
their Wronskian, i.e.,
W
(
y
1
,
y
2
, … ,
y
n
±1
,
y
n
) ≠ 0.
Note 1
:
In order to determine the
n
unknown coefficients
C
i
, each
n
th order
equation requires a set of
n
initial conditions in an initial value problem:
y
(
t
0
) =
y
0
,
y′
(
t
0
) =
y′
0
,
y
″(
t
0
) =
y
″
0
, and
y
(
n
±1)
(
t
0
) =
y
(
n
±1)
0
.
Note 2
:
The Wronskian
W
(
y
1
,
y
2
, … ,
y
n
±1
,
y
n
) is defined to be the
determinant of the following
n
×
n
matrix



)
1
(
)
1
(
2
)
1
(
1
2
1
2
1
2
1
..
..
:
:
:
"
..
..
"
"
'
..
..
'
'
..
..
n
n
n
n
n
n
n
y
y
y
y
y
y
y
y
y
y
y
y
.
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View Full DocumentSuch a set of linearly independent solutions, and therefore, a general solution
of the equation, can be found by first solving the differential equation’s
characteristic equation:
a
n
r
n
+
a
n
1
r
n
1
+ … +
a
2
r
2
+
a
1
r
+
a
0
= 0.
This is a polynomial equation of degree
n
, therefore, it has
n
real and/or
complex roots (not necessarily distinct).
Those necessary
n
linearly
independent solutions can then be found using the four rules below.
(i).
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