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MATH246_COHENJ_FALL2000_0101_MID_EXAM.pdf
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MATH246_COHENJ_FALL2000_0101_MID_EXAM.pdfExam 1 September 26, 2000 Math
MATH246 COHENJ FALL2000 0101 MID E...
MATH246_COHENJ_FALL2000_0101_MID_EXAM.pdfExam 1 September 26, 2000 Math
Page 1
Exam 1
September 26, 2000
Math 246
Joel M. Cohen
3:30  4:45 p.m.
Name:
Read the problems
very
carefully!!
CIRCLE
your ﬁnal answer to each problem!!
When possible, express solutions explicitly.
Show all your work on these pages,
using the backs for scratch paper. Check your answers!
1.
(60 points)
Find the general solutions:
(a)
t
dy
dt
+2
y
=
t
3
.
(b)
t
dy
dt
= (1 +
y
2
)
.
(c)
1
t
+
y

cos
y
+
y
cos
t
¶
dt
+
1
t
+
y
+
t
sin
y
+ sin
t
¶
dy
=0
.
2.
(40 points)
Solve the initial value problems
(a)
dy
dt
+
2
t
y
=
cos
t
t
2
, and
y
(
π
)=0
,t>
0
.
(b) (2
xy

6
x
2
y
2
)
dx
+(
x
2

4
x
3
y
+ 24
y
3
)
dy
=0
,
y
(0) = 1.
3.
(40 points)
I bought 1,000 pounds of radioactive cohenium (used in the produc
tion of a GPA destroying bomb), and had it mailed to me by surface mail. When
it arrived exactly four days later, it weighed 8.1 pounds.
(a) Write down the initial value problem describing this situation, letting
Q
(
t
) rep
resent the quantity of cohenium in the package at time
t
.
(b) Solve the initial value problem and give the precise formula for
Q
(
t
).
(c) If I had paid an extra $1.20 the package would have been delivered in one day.
How much cohenium would there have been then?
4.
(35 points)
Sketch the direction ﬁeld and then some representative integral
curves of this equation, and discuss the stability of the equilibrium solutions:
y
0
=
y
2
(
y

1)(
y

3)
.
5.
(15 points)
A tank initially holds 80 liters of a salt solution containing
1
4
lb. of
salt per liter. Starting at time
t
= 0, a solution containing 1 lb. of salt per liter is
poured into the tank at the rate of 3 liter/min, and a
second
solution containing
1/2 lb. of salt per liter is poured into the tank at the rate of 2 liter/min while the
well–stirred mixture leaves the tank at the rate of 5 liter/min.
Page 2
2
(a)
Set up
the initial value problem for ﬁnding the amount of
salt
in the tank at
any time
t
.
(b)
Solve
the initial value problem.
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