Math 1320 Lab4
Instructor: ridget Fan
Name:
Due: 02,/11/16 in class
Show all work including necessary graphs to get full credits!
1. Use the method of separation of variables to solve the following problems:
(a)
(y + xy)y’
Solve for y, where y’
=
=
1
clx
j(t+?c) i
I Jç
: (1’tfc
(b) Find an equation of the curve that passes through (0, 1) and whose slope at (x, y)
is
6x(y
—
1)2/3.
6?c(
-‘)
f(j :I6,J?c
(-)
3
(zj.C)
j:
(01
) :) 1
3 -k-i
C
co
Page 2 of 4
Lab4
Math 1320
2. A glass of hot water is cooling down with surrounding temperature of 72 degrees. The
rate of change of the water temperature T(I) is proportional to the difference between
T(t) and the surrounding temperature. Suppose at I = 0 the water temperature is 100
degrees and drops to 82 degrees after 10 minutes.
(a) Set up the differential equation describing T(t).
T’ft)
k
(T-2)
(b) Solve the differential equation from part (a). (Hint: Your answer of T(t) should
have two unknowns: the constant of proportionality and the arbitray constant c
from integration.)
(hlT.-i21
(Iq+c)
:e
(c) Use the two
(b).
flz.
conditions given
TeC+12
(j
o)
=
c)
by the problem to solve the two unknowns from part
=) ecz2
too
e° +11
2
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) CIn’I
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)
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e
0
j
1oc
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1
IOW)I(f.
Math 1320
Lab4
Page 3 of 4
3. A population P(t) has constant relative birth and death rates a and , respectively, and
The
a constant emigration rate m (a, and m are positive constants). Assume a >
rate of change of the population at time t is modeled by
.
=
kP
—
m,
where
k
=
a
—
(a) Find the solution of this equation that satisfies the initial condition P(0)
—‘-—dy- cAk
=
Po.
HCc&4v)
ce
(b) What condition on rn relative to kP
0 will lead to exponential expansion of the
population? What condition on m relative to kP
0 will result in constant population?
Population decline?
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Coflsfonl
X ()Oii
OplkAtn
(c) In 1847, the population of Ireland was 8 million and the difference between the
relative birth and death rates was 1.6o. As a result of the potato famine in the
1840s and 1850s, 210, 000 inhabitants per year emigrated from Ireland. Was the
population expanding or declining?
Cr
r ,OVOoOO
000
okx4cw’ 9ec” e..
Page 4 of 4
Lab4
Math 1320
whether the sequence is
4. Determine
(a) a
=
convergent or divergent and explain
why.
e
{ic4
3e(ce
Li
fr,ø
e
e°
:I
11
(b) a
=
Letb fri
JJ—
Je row
( c)
a,
a,, ia,,
&j oqe
i-i
sin(nii)
=
a
3c
(d) a,
—b,
hc
)
fl
Y°flt
=
‘(n)O,
ln(n)
—
ln(n
—
cr:-
1)
—
in(e.
1
COIAU4is
((X) i
..-.)
f.
(e) a,,
=
—
i
( J2J’
/Z 1
-
-
V’-?
-
LO
cWtTh:
—
—
I
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