1000 Solved Problems in Modern Physics

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 LaplaceTransforms

1.72 Consider the chain decay in radioactivityA
λA
→B
λB
→C, whereλAandλBare
the disintegration constants. The equations for the radioactive decays are:
dNA(t)
dt

=−λANA(t),and

dNB(t)

dt

=−λ 2 NB(t)+λANA(t)

whereNA(t) andNB(t) are the number of atoms ofAandBat timet, with

the initial conditionsNA(0)=NA^0 ;NB(0)=0. Apply Laplace transform to

obtainNA(t) andNB(t), the number of atoms ofAandBas a function of time

t, in terms ofN^0 A,λAandλB.

1.73 Consider the radioactive decay:

A

λA

→B

λB

→C(Stable)

The equations for the chain decay are:

dNA

dt

=−λANA (1)

dNB

dt

=−λBNB+λANA (2)

dNC

dt

=+λBNB (3)

with the initial conditionsNA(0)=N^0 A;NB(0)=0;NC(0)=0, where various

symbols have the usual meaning. Apply Laplace transforms to find the growth

ofC.

1.74 Show that:

(a)£(eax)=s−^1 a,if s>a

(b)£(cosax)=s (^2) +sa 2 ,s> 0
(c)£(sinax)=s (^2) +aa 2
where£means Laplacian transform.

1.2.10 Special Functions

1.75 The following polynomial of ordernis called Hermite polynomial:

Hn′′− 2 ξHn′+ 2 nHn= 0

Show that:

(a)Hn′=^2 nHn− 1

(b)Hn+ 1 = 2 ξHn− 2 nHn− 1