1.2 Problems
29
1.71 Find a fundamental set of solutions to the third-order equation:
d
3 y
dx 3 −
d
2 y
dx 2 +
dy
dx
− y = 0
1.2.9 Laplace Transforms
1.72 Consider the chain decay in radioactivity A
λ A
→ B
λ B
→ C, where λ A and λ B are
the disintegration constants. The equations for the radioactive decays are:
dN A (t)
dt
= −λ A N A (t), and
dN B (t)
dt
= −λ 2 N B (t) + λ A N A (t)
where N A (t) and N B (t) are the number of atoms of A and B at time t, with
the initial conditions N A (0) = N
0
A ; N B (0) = 0. Apply Laplace transform to
obtain N A (t) and N B (t), the number of atoms of A and B as a function of time
t, in terms of N
0
A , λ A and λ B .
1.73 Consider the radioactive decay:
A
λ A
→ B
λ B
→ C (Stable)
The equations for the chain decay are:
dN A
dt
= −λ A N A
(1)
dN B
dt
= −λ B N B + λ A N A
(2)
dN C
dt
= +λ B N B
(3)
with the initial conditions N A (0) = N
0
A ; N B (0) = 0; N C (0) = 0, where various
symbols have the usual meaning. Apply Laplace transforms to find the growth
of C.
1.74 Show that:
(a) £(e
ax ) =
1
s−a
, i f s > a
(b) £(cos ax) =
s
s 2 +a 2 , s > 0
(c) £(sin ax) =
a
s 2 +a 2
where £ means Laplacian transform.
1.2.10 Special Functions
1.75 The following polynomial of order n is called Hermite polynomial:
H
n − 2ξ H
n + 2n H n = 0
Show that:
(a) H
n = 2n H n−1
(b) H n+1 = 2ξ H n − 2n H n−1
29
1.71 Find a fundamental set of solutions to the third-order equation:
d
3 y
dx 3 −
d
2 y
dx 2 +
dy
dx
− y = 0
1.2.9 Laplace Transforms
1.72 Consider the chain decay in radioactivity A
λ A
→ B
λ B
→ C, where λ A and λ B are
the disintegration constants. The equations for the radioactive decays are:
dN A (t)
dt
= −λ A N A (t), and
dN B (t)
dt
= −λ 2 N B (t) + λ A N A (t)
where N A (t) and N B (t) are the number of atoms of A and B at time t, with
the initial conditions N A (0) = N
0
A ; N B (0) = 0. Apply Laplace transform to
obtain N A (t) and N B (t), the number of atoms of A and B as a function of time
t, in terms of N
0
A , λ A and λ B .
1.73 Consider the radioactive decay:
A
λ A
→ B
λ B
→ C (Stable)
The equations for the chain decay are:
dN A
dt
= −λ A N A
(1)
dN B
dt
= −λ B N B + λ A N A
(2)
dN C
dt
= +λ B N B
(3)
with the initial conditions N A (0) = N
0
A ; N B (0) = 0; N C (0) = 0, where various
symbols have the usual meaning. Apply Laplace transforms to find the growth
of C.
1.74 Show that:
(a) £(e
ax ) =
1
s−a
, i f s > a
(b) £(cos ax) =
s
s 2 +a 2 , s > 0
(c) £(sin ax) =
a
s 2 +a 2
where £ means Laplacian transform.
1.2.10 Special Functions
1.75 The following polynomial of order n is called Hermite polynomial:
H
n − 2ξ H
n + 2n H n = 0
Show that:
(a) H
n = 2n H n−1
(b) H n+1 = 2ξ H n − 2n H n−1