1000 Solved Problems in Modern Physics

(Romina) #1

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)=N^0 A;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
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