+1 862 207 3288 .
2
1.
A person is trying to fill her new swimming pool for the first time. Water
is flowing into the
pool at a constant rate of 30
0
litres/min. Unfortunately there is a hole and a crack in the pool from
which water is leaking. Water is leaking out of the hole at a rate proportional to the square of the
amount of water currently in th
e pool while water is leaking out of the crack at a rate proportional to
the amount of water currently in the pool. As it had rained the previous day, the pool already
contained 20
0
litres of water before filling began. Let
t
(mins.) be the time since th
e person started
filling the pool and let
W
(
t
)
be the number of litres of water in the pool at time
t
.
Write down, but do not solve,
the differential equation for
W
(
t
)
along with its initial
condition.
2.
Solve the logistic equation
(4 )
dP
PP
dt
= 3 –
with
(0) 2
P
=
.
What value does
P
approach as
t
gets large, ie. as
t
?8
.
3.
(a)
Find the general solution of
2
2
d y dy
y
dt
dt
+ 3 – 4 = 0
.
(b)
Solve
2
2
8cos2 + 6sin2
d y dy
y tt
dt
dt
+ 3 – 4 =
w
ith
yy
(0) = 4, (0) = 0
‘
.
4.
Solve the f
ollowing differential equations
:
(a)
(
)
+ 4 =
dy
x
xy
dx
2
.
(b)
2
2
4 20 0
d y dy
y
dx
dx
++=
.
(c)
–
dy
y
dx
=
.
(d)
2
– 2
dy
yx
dx
=
+
.
(e)
20 100 0
yy y
” ‘
++ =
.
(f)
22
sin
dy
x xy x x
dx
+ =
.
5.
A dice is found to be weighted so that the chance of throwing a 6 is
four
times the chance of
throwing a
1
. Chances of throwing numbers other than 6 are equally likely.
(a)
What is the probability
of throwing a 6?
(b)
What is the probability of throwing a 3?
(c)
What is the probability of throwing at least a 3?
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