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CH3 Modeling with First-Order Differential Equations
3.1 Linear Models
Growth and Decay
The IVP
dx
dt
=kx , x
t
0
=x
0
where k is a constant of proportionality, serve as a model for diverse phenomena
involving either growth or decay.
When k > 0 , k is a growth constant.
When k < 0 , k is a decay constant.
e.g.
A culture initially has P
0
number of bacteria. At t= 1
h the number of bacteria is
measured to be
P
0
. If the rate of growth is proportional to the number of bacteria
P ( t )
present at time t
, determine the time necessary for the number of bacteria to
triple.
dP
dt
−kP=0, P ( 0 )=P
0
, P( 1 ) =
P
0
e
−kt
P=c
P
0
=c
e
−k
P
0
=P
0
k =ln
P
t
=P
0
(
)
t
(
)
t
t=
ln 3
ln
Half-Life:
The time takes for one-half of the atoms in an initial amount A
0
to disintegrate, or
transmute, into the atoms of another element.
e.g.
A breeder reactor converts relatively stable uranium-238 into the isotope plutonium-
- After 15 years it is determined that 0.043% of the initial amount A
0
of
plutonium has disintegrated. Find the half-life of this isotope if the rate of
disintegration is proportional to the amount remaining.
Let A ( t )
denote the amount of plutonium remaining at time t.
dA
dt
=kA , A ( 0 ) =A
0
A
t
= A
0
e
kt
A
= A
0
e
15 k
A
0
k =
ln 0.
A
t
half
A
0
e
k t
half
t
half
ln
k
≈ 24174.35 ( years)
Carbon Dating:
The half-life of C-14 is the Cambridge half-life that is close to 5730 years.
e.g.
A fossilized bone is found to contain 0.1% of its original amount of C-14. Determine
the age of the fossil.
A
t
= A
0
e
kt
A ( 5730 )=
A
0
k =
−ln 2
0.001 A
0
= A
0
e
kt
t=
ln 0.
k
≈ 57103 ( year )
Newton’s Law of Cooling / Warming
dT
dt
=k
T −T
m
k
: constant of proportionality
Logistic Equation:
Carrying capacity: the maximum individuals in an environment
Let K be the carrying capacity.
f ( K )= 0
Let f ( 0 )=r
Simplest assumption for
f ( P)
is linear – that is,
f ( P) =r−
r
K
P
The logistic equation came in the form of
dP
dt
=P ( a−bP)
Solution of the logistic equation:
dP
P ( a−bP )
=dt
1 / a
P
b / a
a−bP
dP=dt
a
ln|P|−
a
ln|a−bP|=t+ c
ln
P
a−bP
=at +ac
P
a−bP
=c
1
e
at
P ( t) =
a c
1
b c
1
+e
−at
If P ( 0 )=P
0
P ( t) =
a P
0
b P
0
a−b P
0
e
−at
Modifications of the logistic equation:
The population in a fishery where fish are harvested or are restocked at rate h.
dP
dt
=P ( a−bP)−h
and
dP
dt
=P ( a−bP) +h
The equation could also serve as models of the human population decreased by
emigration or increased by immigration.
The rate h could be a function of time t or could be population dependent.
Chemical reactions:
Suppose that a grams of chemical A are combined with b grams of chemical B.
mA + nB→ C
and X ( t )
is the number of grams of chemical C
formed.
dX
dt
(
a−
m
m+ n
X
)(
b−
n
m+ n
X
)
3.3 Modeling with Systems of First-Order DEs
