.
.
P
n
=
(1 +
r
Δ
t
)
P
n

1
= (1 +
r
Δ
t
)
n
P
0
The solution of this discrete model is
P
n
= (1 +
r
Δ
t
)
n
P
0
,
which is an exponential growth
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Introduction to Differential Equations
— (15/47)
The Class — Overview
The Class...
Introduction
Applications of Differential Equations
Malthusian Growth
Example
Definitions  What is a Differential Equation?
Classification
Malthusian Growth
3
Discrete Malthusian Growth
:
P
n
+1
= (1 + 0
.
1Δ
t
)
P
n
P
0
= 4
0
2∆
4∆
6∆
8∆
10∆
0
2
4
6
8
10
12
n
P
n
Discrete Malthusian Growth
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Introduction to Differential Equa
— (16/47)
Subscribe to view the full document.
The Class — Overview
The Class...
Introduction
Applications of Differential Equations
Malthusian Growth
Example
Definitions  What is a Differential Equation?
Classification
Malthusian Growth
4
Malthusian Growth
: Let
P
(
t
) be the population at time
t
=
t
0
+
n
∆
t
and rearrange the model above
P
n
+1

P
n
=
r
∆
tP
n
P
(
t
+ ∆
t
)

P
(
t
)
=
∆
t
·
rP
(
t
)
P
(
t
+ ∆
t
)

P
(
t
)
∆
t
=
rP
(
t
)
Let ∆
t
become very small
lim
∆
t
→
0
P
(
t
+ ∆
t
)

P
(
t
)
∆
t
=
dP
(
t
)
dt
=
rP
(
t
)
,
which is a
Differential Equation
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Introduction to Differential Equations
— (17/47)
The Class — Overview
The Class...
Introduction
Applications of Differential Equations
Malthusian Growth
Example
Definitions  What is a Differential Equation?
Classification
Malthusian Growth
5
Solution of Malthusian Growth Model
: The Malthusian
growth model
dP
(
t
)
dt
=
rP
(
t
)
The rate of change of a population is proportional to the
population
Let
c
be an arbitrary constant, so try a solution of the form
P
(
t
) =
ce
kt
Differentiating
dP
(
t
)
dt
=
cke
kt
,
which if
k
=
r
is
rP
(
t
), so satisfies the differential equation
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Introduction to Differential Equa
— (18/47)
The Class — Overview
The Class...
Introduction
Applications of Differential Equations
Malthusian Growth
Example
Definitions  What is a Differential Equation?
Classification
Malthusian Growth
6
Solution of Malthusian Growth Model
The Malthusian
growth model satisfies
P
(
t
) =
ce
rt
With the initial condition,
P
(
t
0
) =
P
0
, then the unique
solution is
P
(
t
) =
P
0
e
r
(
t

t
0
)
Malthusian growth is often called exponential growth
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Introduction to Differential Equations
— (19/47)
The Class — Overview
The Class...
Introduction
Applications of Differential Equations
Malthusian Growth
Example
Definitions  What is a Differential Equation?
Classification
Example: Malthusian Growth
1
Example: Malthusian Growth
Consider the Malthusian
growth model
dP
(
t
)
dt
= 0
.
02
P
(
t
)
with
P
(0) = 100
Skip Example
Find the solution
Determine how long it takes for this population to double
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Introduction to Differential Equa
— (20/47)
The Class — Overview
The Class...
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