Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Separable Differential Equations
— (6/41)
Introduction
Separation of Variables
Modified Malthusian Growth Model
Examples
General Separable Differential Equation
Example of Desiccation of a Cell
1
Desiccation of a Cell:
The model satisfies
dV
dt
=

kV
2
/
3
Suppose that the initial volume of water in the cell is
V
(0) = 8 mm
3
Suppose that 6 hours later the volume of water has
decreased to
V
(6) = 1 mm
3
Solve this differential equation
Find
k
and graph the solution
Determine when all of the water has left the cell
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Separable Differential Equations
— (7/41)
Introduction
Separation of Variables
Modified Malthusian Growth Model
Examples
General Separable Differential Equation
Example of Desiccation of a Cell
2
Solution:
The model is a separable differential equation
dV
dt
=

kV
2
/
3
Separate variables to give
Z
V

2
/
3
dV
=

Z
k dt
Upon integration,
3
V
1
/
3
(
t
) =

kt
+
C
Equivalently,
V
(
t
) =

kt
+
C
3
3
The initial condition gives
V
(0) = 8 =
(
C
3
)
3
or
C
= 6
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Separable Differential Equations
— (8/41)
Subscribe to view the full document.
Introduction
Separation of Variables
Modified Malthusian Growth Model
Examples
General Separable Differential Equation
Example of Desiccation of a Cell
3
Solution:
The model is given by
V
(
t
) =

kt
+ 6
3
3
The other condition gives
V
(6) = 1 =

6
k
+ 6
3
3
= (

2
k
+ 2)
3
So
k
=
1
2
The solution to this problem is
V
(
t
) =
2

t
6
3
The solution vanishes (all the water evaporates) at
t
= 12
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Separable Differential Equations
— (9/41)
Introduction
Separation of Variables
Modified Malthusian Growth Model
Examples
General Separable Differential Equation
Example of Desiccation of a Cell
4
Graphs of Desiccation of a Cell
0
2
4
6
8
10
12
0
2
4
6
8
Time (hr)
Volume (
μ
m
3
)
Dessication of a Cell
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Separable Differential Equations
— (10/41)
Introduction
Separation of Variables
Modified Malthusian Growth Model
Examples
General Separable Differential Equation
Example 1  Separable Differential Equation
1
Example  Separable Differential Equation
Consider the
differential equation
dy
dt
= 2
ty
2
Solution:
Separate the variables
t
and
y
Put only 2
t
and
dt
on the right hand side
And only
y
2
and
dy
are on the left hand side
The integral form is
Z
dy
y
2
=
Z
2
t dt
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Separable Differential Equations
— (11/41)
Introduction
Separation of Variables
Modified Malthusian Growth Model
Examples
General Separable Differential Equation
Example 1  Separable Differential Equation
2
Solution (cont)
The two integrals are
Z
dy
y
2
=
Z
2
t dt
The two integrals are easily solved

1
y
=
t
2
+
C
Note
that you only need to put
one arbitrary constant
,
despite solving two integrals
This is easily rearranged to give the solution in explicit
form
y
(
t
) =

1
t
2
+
C
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Separable Differential Equations
— (12/41)
Introduction
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 Fall '08
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