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Ch 3.5:
Repeated Roots; Reduction of Order
Recall our 2
nd
order linear homogeneous ODE
where
a
,
b
and
c
are constants.
Assuming an exponential soln leads to characteristic equation:
Quadratic formula (or factoring) yields two solutions,
r
1
&
r
2
:
When
b
2
– 4
ac
= 0,
r
1
=
r
2
= 
b
/2
a
, since method only gives
one solution:
0
=
+
′
+
′
′
cy
y
b
y
a
0
)
(
2
=
+
+
⇒
=
c
br
ar
e
t
y
rt
a
ac
b
b
r
2
4
2

±

=
a
t
b
ce
t
y
2
/
1
)
(

=
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Multiplying Factor
v
(
t
)
We know that
Since
y
1
and
y
2
are linearly dependent, we generalize this
approach and multiply by a function
v
, and determine
conditions for which
y
2
is a solution:
Then
solution
a
)
(
)
(
solution
a
)
(
1
2
1
t
cy
t
y
t
y
=
⇒
a
t
b
a
t
b
e
t
v
t
y
e
t
y
2
/
2
2
/
1
)
(
)
(
try
solution
a
)
(


=
⇒
=
a
t
b
a
t
b
a
t
b
a
t
b
a
t
b
a
t
b
a
t
b
e
t
v
a
b
e
t
v
a
b
e
t
v
a
b
e
t
v
t
y
e
t
v
a
b
e
t
v
t
y
e
t
v
t
y
2
/
2
2
2
/
2
/
2
/
2
2
/
2
/
2
2
/
2
)
(
4
)
(
2
)
(
2
)
(
)
(
)
(
2
)
(
)
(
)
(
)
(







+
′

′

′
′
=
′
′

′
=
′
=
Finding Multiplying Factor
v
(
t
)
Substituting derivatives into ODE, we seek a formula for
v
:
0
=
+
′
+
′
′
cy
y
b
y
a
4
3
2
2
2
2
2
2
2
2
2
2
2
/
)
(
0
)
(
0
)
(
4
4
)
(
0
)
(
4
4
4
)
(
0
)
(
4
4
4
2
4
)
(
0
)
(
2
4
)
(
0
)
(
)
(
2
)
(
)
(
4
)
(
)
(
0
)
(
)
(
2
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This note was uploaded on 08/13/2009 for the course DIFF 2343632 taught by Professor Csar during the Fall '09 term at Middle East Technical University.
 Fall '09
 Csar

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