### 8.8.3. Case II: Real Roots, Some Repeated

If all roots are real but not distinct,
then the *u*_{k}’s
in (8.14) are not independent so that the
general solution (8.13) needs to be accordingly modified.

#### Theorem 8.9.

The *m*
functions

u
k
(
x
)=
x
k−1
e
rx
with
k=1,⋯,m

are *m*
independent
elements annihilated by
(
D−r
)
m
.

##### Proof

Independence of the *u*_{k}’s
follows from that of the monomials
x
k−1
. The proof that each *u*_{k}
is annihilated by
(
D−r
)
m
is by induction
on *m*.
The case
m=1
is trivial.
Thus, assuming case *m*-1 holds, we wish
to prove case *m*. To be more specific, we assume the functions
u
1
,⋯,
u
m−1
are all annihilated by
(
D−r
)
m−1
. Now,

(
D−r
)
u
m
=(
D−r
)(
x
m−1
e
rx
)
=(
m−1
)
x
m−2
e
rx
=(
m−1
)
u
m−1

Hence,

(
D−r
)
m
u
m
=(
m−1
)
(
D−r
)
m−1
u
m−1
=0

where the induction hypothesis was used in the last
equality. QED.

#### Example 2.

Find the general solution of
L(
y
)=0
,
where

L=
D
3
−
D
2
−8D+12

##### Solution

L=
(
D−2
)
2
(
D+3
)

Hence,

N(
D−2
)=ℒ{
e
2x
,x
e
2x
}
and
N(
D+3
)=ℒ{
e
−3x
}

so that the general solution is

y=
c
1
e
2x
+
c
2
x
e
2x
+
c
3
e
−3x

#### Example 3.

Solve
(
D
6
+2
D
5
−2
D
3
−
D
2
)y=0
.

##### Solution

From

D
6
+2
D
5
−2
D
3
−
D
2
=
D
2
(
D−1
)
(
D+1
)
3

we get

y=
c
1
+
c
2
x+
c
3
e
x
+(
c
4
+
c
5
x+
c
6
x
2
)
e
−x