1529
SECTION 15.4
.
.
Power Series Solutions of Differential Equations
1249
glucose/100 ml blood two hours after the injection and 78 mg
glucose/100 ml blood three hours after the injection.
40.
Show that the data in exercise 39 are inconsistent with the case
0
< ω < α
.
41.
Consider an
RLC
circuit with capacitance
C
and charge
Q
(
t
)
at time
t
. The energy in the circuit at time
t
is given by
u
(
t
)
=
[
Q
(
t
)]
2
2
C
. Show that the charge in a general
RLC
circuit
has the form
Q
(
t
)
=
e
−
(
R
/
L
)
t
/
2

Q
0
cos
ω
t
+
c
2
sin
ω
t

,
where
Q
0
=
Q
(0) and
ω
=
1
2
L
R
2
−
4
L
/
C
. The relative
energy loss from time
t
=
0 to time
t
=
2
π
ω
is given by
U
loss
=
u
(2
π/ω
)
−
u
(0)
u
(0)
and the
inductance quality factor
is defined by
2
π
U
loss
. Using a Taylor polynomial approximation
of
e
x
, show that the inductance quality factor is approximately
ω
L
R
.
EXPLORATORY EXERCISES
1.
In quantum mechanics, the possible locations of a particle are
described by its
wave function
(
x
). The wave function sat
isfies
Schr¨odinger’s wave equation
¯
h
2
m
(
x
)
+
V
(
x
)
(
x
)
=
E
(
x
)
.
Here, ¯
h
is Planck’s constant,
m
is mass,
V
(
x
) is the potential
function for external forces and
E
is the particle’s energy. In
the case of a bound particle with an infinite square well of
width 2
a
, the potential function is
V
(
x
)
=
0 for
−
a
≤
x
≤
a
.
We will show that the particle’s energy is quantized by solv
ing the
boundary value problem
consisting of the differential
equation
¯
h
2
m
(
x
)
+
v
(
x
)
(
x
)
=
E
(
x
) plus the boundary
conditions
(
−
a
)
=
0 and
(
a
)
=
0. The theory of boundary
value problems is different from that of the initial value prob
lems in this chapter, which typically have unique solutions.
In fact, in this exercise we specifically want more than one
solution. Start with the differential equation and show that for
V
(
x
)
=
0; the general solution is
(
x
)
=
c
1
cos
kx
+
c
2
sin
kx
,
where
k
=
√
2
mE
/
¯
h
. Then set up the equations
(
−
a
)
=
0
and
(
a
)
=
0. Both equations are true if
c
1
=
c
2
=
0, but in
this case the solution would be
(
x
)
=
0. To find
nontrivial
solutions
(that is, nonzero solutions), find all values of
k
such
that cos
ka
=
0 or sin
ka
=
0. Then, solve for the energy
E
in
terms of
a, m
and ¯
h
. These are the only allowable energy levels
for the particle. Finally, determine what happens to the energy
levels as
a
increases without bound.
2.
Imagine a hole drilled through the center of the Earth. What
would happen to a ball dropped in the hole? Galileo conjec
tured that the ball would undergo
simple harmonic motion,
which is the periodic motion of an undamped spring or pendu
lum. This solution requires no friction and a nonrotating Earth.
The force due to gravity of two objects
r
units apart is
Gm
1
m
2
r
2
,
where
G
is the universal gravitation constant and
m
1
and
m
2
are the masses of the objects. Let
R
be the radius of the Earth
and
y
the displacement from the center of the Earth.
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 Spring '10
 CARTER
 Differential Equations, Power Series, Taylor Series, Boundary value problem

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