1000 Solved Problems in Modern Physics

1.3 Solutions 67

1.3.9 LaplaceTransforms

1.72

dNA(t)

dt

=−λANA(t)(1)

dNB(t)

dt

=−λBNB(t)+λANA(t)(2)

Applying Laplace transform to (1)

sL(NA)−NA(0)=−λAL(NA)

orL(NA)=

N^0 A

s+λA

=

N^0 A

s−(−λA)

(3)

∴NA=N^0 Aexp(–λAt)(4)

Applying the Laplace transform to (2)

sL(NB)−NB(0)=−λBL(NB)+λAL(NA)(5)

Using (3) in (4) and puttingN 2 (0)= 0

L(NB)(s+λB)=

λAN^0 A

s+λA

orL(NB)=

λAN^0 A

(s+λA)(s+λB)

=

λANA^0

λB−λA

[

1

s+λA

−

1

s+λB

]

=

λANA^0

λB−λA

[

1

s−(−λA)

−

1

s−(−λB)

]

∴NB=

λAN^0 A

λB−λA

[

e−λAt−e−λBt

]

1.73

dNA

dt

=−λANA (1)

dNB

dt

=−λBNB+λANA (2)

dNC

dt

=+λBNB (3)

Applying the Laplace transform to (3)

sL{NC}−NC(0)=λBL{NB}=

λBλAN^0 A

(s+λA)(s+λB)

GivenNc(0)= 0

L{Nc}=

λAλBN^0 A

s(s+λA)(s+λB)

=

λAλBN^0 A

(λB−λA)s

[

1

s+λA

−

1

s+λB

]