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equation relating the diode current and voltage, namely
i
=
(
V
0
−
V
D
)
/
R
.
(2)
Notenowthat(2)saysthat
i
isadecreasinglinearfunctionof
V
D
withvalue
V
0
/
R
when
V
D
is zero. At the same time (1) says that
i
is an exponentially
growing function of
V
D
starting out at
I
0
. Since typically,
RI
0
<
V
0
, the two
resulting curves (for
i
as a function of
V
D
) must cross once. Eliminating
i
from the two equations, we see that the voltage in the diode must satisfy
the transcendental equation
(
V
0
−
V
D
)
/
R
=
I
0
exp(
V
D
/
V
T
)
,
or
V
D
=
V
0
−
RI
0
exp(
V
D
/
V
T
)
.
(a) Reasonable values for the electrical constants are:
V
0
=
1
.
5 volts,
R
=
1000ohms,
I
0
=
10
−
5
amperes,and
V
T
=
.
0025volts.Use
fzero
to find the voltage
V
D
and current
i
in the circuit.
(b) In the remainder of the problem, we assume the voltage in the bat-
tery
V
0
and the resistance of the resistor
R
are unchanged. But
suppose we have some freedom to alter the electrical characteristics
of the diode. For example, suppose that
I
0
is halved. What happens
to the voltage?
(c) Suppose instead of halving
I
0
, we halve
V
T
. Then what is the effect
on
V
D
?
(d) Suppose both
I
0
and
V
T
are cut in half. What then?
(e) Finally,wewanttoexaminethebehaviorofthevoltageifboth
I
0
and
V
T
aredecreasedtowardzero.Fordefinitiveness,assumethatweset
I
0
=
10
−
5
u
and
V
T
=
.
0025
u
, and let
u
→
0. Specifically, compute the
solution for
u
=
10
−
j
,
j
=
0
,...,
5. Then, display a
loglog
plot of
the solution values, for the voltage as a function of
I
0
. What do you
conclude?
11. This problem is based on both the
Population Dynamics
and
360˚ Pendu-
lum
applications from Chapter 9. The growth of a species was modeled in
the former by a
difference equation
. In this problem we will model pop-
ulation growthby a
differential equation
, akin to the second application
mentioned above. In fact we can give a differential equation model for the
logistic growthof a population
x
as a function of time
t
by the equation
x
=
x
(1
−
x
)
=
x
−
x
2
(3)
,
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