1
Differential Equations of Growth
Differential Equations of Growth
dy
D cy
dt
Complete solution
y.t/ D Ae ct
Starting from y.0/ y.t/ D y.0/e
Now include a constant source term s
dy
D cy C s
dt
for any A
ct
A D y.0/
This gives a new equation
s � 0 is saving, s � 0 is spending, cy is interest
s
Complete solution y.t/ D � C Ae ct (any A gives a solution)
c
s
dy
D 0 and A D 0
y D � is a constant solution with cy C s D 0 and
c
dt
For that solution, the spending s exactly balances the income cy
s s ct
Choose A to start from y.0/ at t D 0 y.t/ D � C y.0/ C
e
c
c
Now add a nonlinear term sP 2 coming from competition
P .t/ D world population at time t (for example) follows a new equation
dP
D cP � sP 2 c D birth rate minus death rate
dt
P 2 since each person competes with each person
1
To bring back a linear equation set y D
P
c
dy
dP =dt
.�cP C sP 2 /
D
D � C s D �cy C s
Then
D�
2
2
P
P
dt
P
“LOGISTIC EQN”
y D 1=P produced our linear equation (no y 2 ) with �c not Cc
s
s s
y.t/ D C Ae �ct D C y.0/ � e �ct D old solution with change to �c
c
c
c
At t D 0 we correctly get y.0/ CORRECT START
As t � � and e �ct � 0 we get y.�/ D
s
c
and P .�/ D
c
s
The population P .t/ increases along an S -curve approaching
c
s
2
Differential Equations of Growth
c
has P � D 0 Inflection point Bending changes from up to down
2s
d
dP
c
d 2P
D 0 at P D
CHECK
D
cP � sP 2 D .c � 2sP /
2
dt
dt
2s
dt
c
World population approaches the limit � 12 billion (FOR THIS MODEL!)
s
Population now � 7 billion Try Google for “World population”
PD
Practice Questions
dy
D cy � s has s D spending rate not savings rate (with minus sign)
dt
dy
1. The constant solution is y D
when
D0
dt
In that case interest income balances spending: cy D s
2. The complete solution is y.t/ D
s
s
C Ae ct : Why is A D y.0/ � ?
c
c
3. If you start with y.0/ �
s
why does wealth approach � ?
c
s
If you start with y.0/ � why does wealth approach �� ?
c
4. The complete solution to
dy
D s is y.t/ D st C A
dt
What solution y.t/ starts from y.0/ at t D 0 ?
5. If
dP
1
dy
explain why
Ds
D �sP 2 and y D
dt
P
dt
Pure competition. Show that P .t/ � 0 as t � �
6. If
dP
1
D cP � sP 4 find a linear equation for y D 3
dt
P
MIT OpenCourseWare
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Resource: Highlights of Calculus
Gilbert Strang
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