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Summer 2012
~*S. ~
MATH 2250-001: Exam 1
~M~- ~s
Name:
~
~C
______________
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: Identify the following differential equations as one of the following (Note:
DO NOT solve any of the differential equations):
a.) First-Order Linear Differential Equation
b.) Separable Differential Equation
c.) Second-Order Differential Equation
d.) Autonomous Differential Equation
,~,
L~X2—4X+4
n~
2.
+ sin(x)y
~
cos(x)
-
Ov~
~cu~
~
2~ 3.15y”—lOxy’=y
‘S
4.
tan(x)~
=
___
\)~jc~xt~c&
1
Efr~cd~
~Opl-s
Problem 2: Find a general solution to the ODE
dy
dx
x— + y
=
3xy
—b
±
‘5
4
c~
~
C
)cQ
4-
a
ax
-35<
(~—gy~’)
j
C
2
-3X
~=0
Problem 3: A 120-gallon tank initially contains 90 pounds of salt dissolved in 90 gallons
of water. Brine containing 2 pounds per gallon of salt flows into the tank at the rate of 4
gallons per minute, and the well-stirred mixture flows out of the tank at the rate of 3 gallons
per minute. How much salt does the tank contain when it is full?
V(oN~ ‘73
XtoN
~
9 Cc~
-
So
______
~°tt
0
dt
~r-e~-
~C0
‘~
0
-3
‘A
3
9ot-&
r/if~
-,
~~1
3
3
(7of&~
(~o*t~
Kit~)
%l~N~
(7oF&~
~f
V i-c
ylo\~ 70
if
2(10* tN
-tE~- 3o
M~i..
Problem 4: All parts of this problem deal with the following differential equation:
dx
2
—=9—x
dt
a.) Find the critical points of the ODE
<:-=~)
(3—)(~1-x~)~c~
b.) Determinc whcthcr the critical points found in part a.) are stable, unstable, or
neither by drawing the phase diagram.
ckt
>0
C,
<0
co
~t-3f3
~
0
S.
c.) Solve the above ODE for x(t)
at
(3-
;~(3+-s~N
/~
(
)
(3Ha(3b~
(3-xN(2 Ha
c::. ~
c;zo
Ii
÷
4
~! ~
-
Co
2*
(0tf C
tV~~ a~,coktLj-t
4
‘at
Problem 5: Solve the following homogeneous linear system and write the solution in
vector form.
—
x2 + x3 + 7x4 + 3x5
=
0
2z5
=
0
—
(
j
~O
C
tf
\-i-z
L
0
i
—
0
—
0
0
H -a
%\-
0
—s~
t-~r
X3=
519
H
xs
—
±
a
-F
—21
01
“C
0
1/4-
0
0,-I
5
r-s
&
0
2.
7
Lb ?*S
Problem 6: Let A
=
2
—1
—1 —1
2 —1
—1
—1
Compute det(A).
2
+ (-~
&
6
~ç3~
Extra CreditS: If A is an n x n matrix and A2
~ E~N
=
A, show that det(A)
=
~ ~
cc,
~
c~’
4?
7
0 or 1.