= 0 upstream,
then a reasonable model for its concentration in the river is
p
(
t
) = 8
e

0
.
002
t
ppm (parts per million)
Form the differential equation describing the concentration of
pollutant in the lake at any time
t
and solve it
Graph the solution and approximate how long it takes for this
lake to have a concentration of 2 ppm
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Linear Differential Equations
— (32/64)
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Introduction
Falling Cat
1
st
Order Linear DEs
Examples
Pollution in a Lake
Example 2
Mercury in Fish
Modeling Mercury in Fish
Example 2: Pollution in a Lake
2
Solution:
This model follows the original derivation above with
V
= 10
5
,
f
(
t
) = 100 + 60 sin(0
.
0172
t
), and
p
(
t
) = 8
e

0
.
002
t
, so the DE
for the concentration of pollutant is
dc
(
t
)
dt
=

f
(
t
)
V
(
c
(
t
)

p
(
t
))
with
c
(0) = 0
=

(0
.
001 + 0
.
0006 sin(0
.
0172
t
))
(
c
(
t
)

8
e

0
.
002
t
)
This requires use of an integrating factor
μ
(
t
) =
e
R
(0
.
001+0
.
0006 sin(0
.
0172
t
))
dt
=
e
0
.
001
t

0
.
0349 cos(0
.
0172
t
)
so
d
dt
e
0
.
001
t

0
.
0349 cos(0
.
0172
t
)
c
(
t
)
= (0
.
008+0
.
0048 sin(0
.
0172
t
))
e

0
.
001
t

0
.
0349 cos(0
.
0172
t
)
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Linear Differential Equations
— (33/64)
Introduction
Falling Cat
1
st
Order Linear DEs
Examples
Pollution in a Lake
Example 2
Mercury in Fish
Modeling Mercury in Fish
Example 2: Pollution in a Lake
3
Solution:
From before,
d
dt
e
0
.
001
t

0
.
0349 cos(0
.
0172
t
)
c
(
t
)
= (0
.
008 + 0
.
0048 sin(0
.
0172
t
))
e

0
.
001
t

0
.
0349 cos(0
.
0172
t
)
The integrating gives
e
0
.
001
t

0
.
0349 cos(0
.
0172
t
)
c
(
t
) =
Z
(0
.
008 + 0
.
0048 sin (0
.
0172
t
)) e

0
.
001
t

0
.
0349 cos(0
.
0172
t
)
dt.
This last integral cannot be solved, even with
Maple
.
Numerical methods are needed to solve and graph this problem, and
our preferred method is MatLab
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Linear Differential Equations
— (34/64)
Introduction
Falling Cat
1
st
Order Linear DEs
Examples
Pollution in a Lake
Example 2
Mercury in Fish
Modeling Mercury in Fish
Example 2: Pollution in a Lake
4
MatLab Solution:
The pollution problem is integrated numerically
(ode23). MatLab finds that the pollution exceeds 2 ppm after
t
= 447
.
4 days
. Below shows a graph. (Programs are provided on
Lecture
page
.)
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0
0.5
1
1.5
2
2.5
3
Critical time
t
days
c
(
t
) ppm
Lake Pollution
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Linear Differential Equations
— (35/64)
Introduction
Falling Cat
1
st
Order Linear DEs
Examples
Pollution in a Lake
Example 2
Mercury in Fish
Modeling Mercury in Fish
Pollution in a Lake: Complications
Pollution in a Lake: Complications
The above examples for
pollution in a lake fail to account for many significant complications
There are considerations of irregular variations of pollutant
entering, stratification in the lake, and uptake and reentering of
the pollutant through interaction with the organisms living in
the lake
The river will have varying flow rates, and the leeching of the
pollutant into river is highly dependent on rainfall, ground water
movement, and rate of pollutant introduction
Obviously, there are many other complications that would
increase the difficulty of analyzing this model
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Linear Differential Equations
— (36/64)
Introduction
Falling Cat
1
st
Order Linear DEs
Examples
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