AMS 361: Applied Calculus IV by Prof Y. Deng
Homework 6
Assignment Date:
Wednesday (03/18/200
9
)
Collection Date:
Wednesday (04/01/2009)
No HW Due 3/25 (Quiz2 Date)
Grade:
Each problem is worth 10 points.
Problem 6.1
Find the general solution of the following equation
ሺܦ െ 4ሻ
ସ
ሺܦ െ 2ሻ
ଶ
ሺܦ
ଶ
െ 2ܦ െ 3ሻ ൌ 0 where ܦ ؠ
݀
݀ݔ
Solution:
The characteristic equation is
.
0
)
3
2
(
)
2
(
)
4
(
2
2
4
=
−
−
−
−
r
r
r
r
We have
.
1
,
3
,
2
,
4
8
7
6
5
4
3
2
1
−
=
=
=
=
=
=
=
=
r
r
r
r
r
r
r
r
Therefore the general solution is
.
)
(
)
(
)
(
8
3
7
2
6
5
4
3
4
2
3
2
1
x
x
x
x
e
c
e
c
e
x
c
c
e
x
c
x
c
x
c
c
x
y
−
+
+
+
+
+
+
+
=
Problem 6.2
Find the general solution of the following equation
൬ݔ
݀
݀ݔ
െ ߙ൰
ݕሺݔሻ ൌ 0 where ߙ ൌ constant and ݊ ൌ integer
Solution:
Let
,
ln
x
r
=
),
(
)
(
r
e
y
r
u
=
we have
dr
d
x
dx
dr
dr
d
dx
d
1
=
⋅
=
,
.
1
α
α
α
−
=
−
⋅
=
−
dr
d
dr
d
x
x
dx
d
x
.
0
)
(
)
(
)
(
)
(
)
(
)
(
=
−
=
−
=
−
r
u
dr
d
e
y
dr
d
x
y
dx
d
x
n
r
n
n
α
α
α
Then
.
...
2
1
α
=
=
=
=
n
r
r
r
We have
.
)
...
(
)
(
1
1
2
1
r
n
n
e
r
c
r
c
c
r
u
α
−
−
+
+
+
=
.
)
...
(
)
(
1
1
2
1
r
n
n
r
e
r
c
r
c
c
e
y
α
−
−
+
+
+
=
Then the general solution is
{
.
}
)
(ln
...
ln
)
(
1
1
2
1
α
x
x
c
x
c
c
x
y
n
n
−
−
+
+
+
=
Problem 6.3
(Prob. 34, P. 168 Modified)
Compute the following Vandermonde

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*