Solutions to Assignment 2  MAT1332A  Fall 2009
(1)
(a) [6 points]
Check that
N
(
t
) =
t
1 +
ct
is a solution of the differential
equation
dN
dt
=
N
2
t
2
. Treat
c
as an unspecified constant.
(b) [5 points]
Use
N
(1) =

1 to find
c
. Then give the solution
N
(
t
) corre
sponding to this initial condition.
Solution: (a)
On one hand,
dN
dt
=
d
dt
t
1 +
ct
=
1 +
ct

ct
(1 +
ct
)
2
=
1
(1 +
ct
)
2
,
and on the other hand,
N
2
t
2
=
t
2
(1 +
ct
)
2
·
1
t
2
=
1
(1 +
ct
)
2
.
Hence
dN
dt
=
N
2
t
2
.
(b)
N
(1) =
1
1 +
c
=

1 =
⇒ 
1

c
= 1 =
⇒
c
=

2
Therefore
N
(
t
) =
t
1

2
t
.
(2)
The rate at which a bacteria population multiplies is proportional to
the instantaneous amount of bacteria present at any time. The mathematical
model for this dynamics can be formulated as follows:
db
dt
=
kb,
where
b
is a function in terms of the time
t
,
b
(
t
) is the number of bacteria at the
time
t
, and
k
is a constant. The general solution of this autonomous differential
equation is
b
(
t
) =
b
(0)
e
kt
,
as shown in class.
(a) [5 points]
Given that the initial population size
b
(0) doubles in two hours,
find
k
. (
Hint: use the logarithmic function
)
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 Fall '09
 ARIANEMASUDA
 one hand, 2k, two hours, four hours

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