Summer 2012
MATH 2250-001: Exam 2
Robbie Sneliman
Instructions: The following questions are intended to assess your abilities on the basic concepts that we
have covered thus far. Answer all questions to the best of your ability and simplify all solutions as much as
possible. Regardless of the simplicity of the problem all work must be shown in order to receive full credit,
otherwise no credit will be awarded.
~
Problem 1: Write the vector
(—1,—2,2) andzD= (4,4,1).
U~L
£WSJ
v3a’q4
F
=
(0,0,19) as a linear combination of the vectors ~
~2- c
&\\3,CE
O%~
~
~C.)
cc
(1,4,3), ~
=
U.bk
ji~*~k vn€~.~ x
c
/
(
—S
~
-2
H
°
~
~-qe~
~
1
0
i
—12.?
—
5 —n(i
~
i
—t
0
--b)
~
~
I ~
~
-~
~
-
=
/
is
1
0
01
~
I
°
II
&~a,6t~1c~j
~)
a(1\
ç)
c~%
/ Lfl
I
~q)(
1
a~c~~t
O\
~
~
(
—‘
C
~
—‘
I
—lii
it,)
Problem 2:
• Let T4~
=
{(x, y, z) e K3
2w + 3y + 4z
LA)
=
ft a..
~a’.cCp,ctUL
such that
&~
~
Ccc~i-ant.s
+
~2
=
~‘-tcinc
~
U~
‘~
0
~
a
~
e K4 w1 +
{(x1,z2, x~, $4)
~i
1} (i.e. T’V is the set of vectors (wy, z) in K3 such that
~
£4
=
=
1). Determine whether H~ is a subspace of K3 or not. All work must be justified.
~
Let W~
2w + 3y + 4z
x3 +
$4).
~2
=
x~ +
x4}
(i.e. 14~ is the set of vectors (zr, X2, $3, $4) in K4
Determine whether W is a subspace of K4 or not. All work must be
justified.
~
~
~
~
~&
C.CCt~~
Ce~ ~A
U(~cUr
~
CV~ 4C~
ux~&~x
~
CXt~CWq
w~J~-~’p icc
cch~vx
~
0.
2
cc
~
La
~
ftC(oce~
Problem 3:
çptc.
•
Find a general solution to the following differential equation:
5y(4) +
=
0
~$=o
(r~
ft
c~
~jj-ecunJ2
• Find a general solution to the following differential equation:
=
e~.
161/
-
~ct-
4N(r~q~=o
~
C=~)
c~
‘C~—2~2,2t,~Li.
~
3
~ ~
c&L4-~,~~
Problem 4: Find Yc and
then write a general solution to the following differential equation:
VP
+ 4y’
=
3x
—
1
Q*~cuc e~.
~(
c~
Co
~‘t~N~
CCos(z~-N
4
~o~Z
—~
~-
~
ai~’
(~
~
co ~o
4- ~
7—pc
(~N
(t~j
(‘~
-=
xtLt~x_~l
~
1~a
8
&-)
n
4
a
±
~ çcin(2~
4
4 ~
x~—
1s~c
Problem 5: Find the inverse Laplace transform for
F(s)
—
—
3—lOs
82_16
You must show your work to
receive full credit on this problem.
4—
~
~q
k\-~~~L
s_loS
c3
stdc
—
T
~
4S~ —~3 (~S
~
~ 1
c~-I(o
-
~
I
5
~
Extra Crediva Find a linear homogeneous constant-coefficient differential equation with the follow
ing general solution:
y(z)
ck
~
±iL
=
(A + Bx + Ox2) cos(2x) + (V + Ex + Fx2) sin(2x)
c\wcccfrcsVtJ,’c
-~
~
&
(N-
2~
~
t4)
to
~
~
4
32Y2*
~L(
to
~
~ o~ e.
6