MATH 251.6/8/11/12
Problem set #1
Due: 9302010
1.
D’oh!
Yet again Homer Simpson has had one
Duff
beer too many before work.
As
a result, water containing 5 kg/m
3
of radioactive plutonium244 waste is flowing into
Springfield’s reservoir at a rate of 4 m
3
per minute.
The reservoir initially contains 2000
m
3
of fresh water.
The mixture (assuming uniform concentration) is drawn off at a rate
of 2 m
3
per minute.
Find an expression for the concentration of radioactive material in
the reservoir at time
t
.
2.
The process of radioactive decay is described by the equation
y
′ = –
r
y
, where
r
is a
positive constant (the
decay constant
of the radioactive material).
Given that Plutonium
244 has a halflife of 8
×
10
7
years, find its decay constant
r
by first solving the initial
value problem:
y
′ = –
r
y
,
y
(0) =
y
0
(where
y
0
> 0).
Then use the half
life to find
r
.
3.
Consider the autonomous equation
y
′ =
y
3
(
y
– 1)
2
(3
y
± 24).
(a) Find all equilibrium solution(s), and classify the stability of each equilibrium solution.
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 Spring '08
 CHEZHONGYUAN
 Math, Differential Equations, Equations, Partial Differential Equations, Radioactive Decay, Homer Simpson, 5 kg

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