5.
(20 points) The linear system
x
1
+
x
2
+ 3
x
3
= 11

x
1
+
x
2
+
x
3
= 11
x
1

x
2

x
3
= 0
x
2
+ 2
x
3
= 11
4
is inconsistent. Find the normal equations that determine a least squares solution to
this system. (Do not solve them.)
This linear system can be written as
A~x
=
~
b
, where
A
=
1
1
3

1
1
1
1

1

1
0
1
2
and
~
b
=
11
11
0
11
.
The normal equations are
A
T
A~x
=
A
T
~
b
. We have
A
T
A
=
1

1
1
0
1
1

1
1
3
1

1
2
1
1
3

1
1
1
1

1

1
0
1
2
=
3

1
1

1
4
7
1
7
15
and
A
T
~
b
=
1

1
1
0
1
1

1
1
3
1

1
2
11
11
0
11
=
0
33
66
.
Therefore the normal equations are
3
x
1

x
2
+
x
3
= 0

x
1
+ 4
x
2
+ 7
x
3
= 33
x
1
+ 7
x
2
+ 15
x
3
= 66
.
6.
(25 points) Find a general solution to the differential equation
y
00

2
y
0
+
y
= 4
e
t
+ 3
t .
The characteristic polynomial is
r
2

2
r
+ 1 = (
r

1)
2
. A general solution to the
associated homogeneous problem is
c
1
e
t
+
c
2
te
t
. Using the Method of Undetermined
Coefficients to find a particular solution
y
p
involves a trial solution of the form
y
p
=
At
2
e
t
+
Bt
+
C .
One has
y
0
p
= 2
Ate
t
+
At
2
e
t
+
B
y
00
p
= 2
Ae
t
+ 4
Ate
t
+
At
2
e
t
.
5
Plugging this into the lefthand side of the differential equation gives
y
00
p

2
y
0
p
+
y
p
= 2
Ae
t
+ 4
Ate
t
+
At
2
e
t

4
Ate
t

2
At
2
e
5

2
B
+
At
2
e
t
+
Bt
+
C
= 2
Ae
t
+
Bt
+ (
C

2
B
)
.
Setting this equal to 4
e
t
+ 3
t
and equating coefficients gives 2
A
= 4 ,
B
= 3 , and
C

2
B
= 0 ; therefore
A
= 2 ,
B
= 3 , and
C
= 6 . This gives
y
p
= 2
t
2
e
t
+ 3
t
+ 6 , so
a general solution to the nonhomogeneous problem is
y
= 2
t
2
e
t
+ 3
t
+ 6 +
c
1
e
t
+
c
2
te
t
.
7.
(16 points) Express the system
x
00
+ 3
x
0
+ 2
x
+ 7
y
=
e
t
y
0
+
x
0
+
x

y
= cos
t
x
(0) = 3
, x
0
(0) = 5
, y
(0) =

1
as a matrix system in the form
~x
0
=
A~x
+
~
f
,
~x
(0) =
~x
0
. (Do not solve the system.)
With the assignments in the lefthand column and the derivatives in the right:
x
1
=
x
x
0
1
=
x
0
=
x
2
x
2
=
x
0
x
0
2
=
x
00
=
e
t

3
x
0

2
x

7
y
=
e
t

3
x
2

2
x
1

7
x
3
x
3
=
y
x
0
3
=
y
0
= cos
t

x
0

x
+
y
= cos
t

x
2

x
1
+
You've reached the end of your free preview.
Want to read all 7 pages?
 Spring '08
 Chorin
 Math, Linear Algebra, Cos, −1, yp