7
Ordinary di
ff
erential equations
A di
ff
erential equation is an equation involving one or more derivatives
of an unknown function. If all derivatives are taken with respect to a
single independent variable we call it an
ordinary di
ff
erential equation
,
whereas we have a
partial di
ff
erential equation
when partial derivatives
are present.
The di
ff
erential equation (ordinary or partial) has
order p
if
p
is the
maximum order of di
ff
erentiation that is present. The next chapter will
be devoted to the study of partial di
ff
erential equations, whereas in the
present chapter we will deal with ordinary di
ff
erential equations of first
order.
Ordinary di
ff
erential equations describe the evolution of many phe
nomena in various fields, as we can see from the following four examples.
Problem 7.1 (Thermodynamics)
Consider a body having internal
temperature
T
which is set in an environment with constant temperature
T
e
. Assume that its mass
m
is concentrated in a single point. Then the
heat transfer between the body and the external environment can be
described by the StefanBoltzmann law
v
(
t
) =
γ
S
(
T
4
(
t
)

T
4
e
)
,
where
t
is the time variable,
the Boltzmann constant (equal to 5
.
6
·
10

8
J
/
m
2
K
4
s where J stands for Joule, K for Kelvin and, obviously, m
for meter, s for second),
γ
is the emissivity constant of the body,
S
the
area of its surface and
v
is the rate of the heat transfer. The rate of
variation of the energy
E
(
t
) =
mCT
(
t
) (where
C
denotes the specific
heat of the material constituting the body) equals, in absolute value,
the rate
v
. Consequently, setting
T
(0) =
T
0
, the computation of
T
(
t
)
requires the solution of the ordinary di
ff
erential equation
dT
dt
=

v
(
t
)
mC
.
(7.1)
188
7 Ordinary di
ff
erential equations
See Exercise 7.15.
Problem 7.2 (Population dynamics)
Consider a population of bac
teria in a confined environment in which no more than
B
elements can
coexist. Assume that, at the initial time, the number of individuals is
equal to
y
0
B
and the growth rate of the bacteria is a positive con
stant
C
. In this case the rate of change of the population is proportional
to the number of existing bacteria, under the restriction that the total
number cannot exceed
B
. This is expressed by the di
ff
erential equation
d
y
d
t
=
Cy
1

y
B
,
(7.2)
whose solution
y
=
y
(
t
) denotes the number of bacteria at time
t
.
Assuming that two populations
y
1
and
y
2
be in competition, instead
of (7.2) we would have
d
y
1
d
t
=
C
1
y
1
(1

b
1
y
1

d
2
y
2
)
,
d
y
2
d
t
=

C
2
y
2
(1

b
2
y
2

d
1
y
1
)
,
(7.3)
where
C
1
and
C
2
represent the growth rates of the two populations.
The coe
ffi
cients
d
1
and
d
2
govern the type of interaction between the
two populations, while
b
1
and
b
2
are related to the available quantity
of nutrients. The above equations (7.3) are called the LotkaVolterra
equations and form the basis of various applications. For their numerical
solution, see Example 7.7.