Mathematical Methods for Physics and Engineering : A Comprehensive Guide

15.4 EXERCISES

15.9 Find the general solutions of

(a)

d^3 y

dx^3

− 12

dy

dx

+16y=32x− 8 ,

(b)

d

dx

(

1

y

dy

dx

)

+(2acoth 2ax)

(

1

y

dy

dx

)

=2a^2 ,

whereais a constant.
15.10 Use the method of Laplace transforms to solve

(a)

d^2 f

dt^2

+5

df

dt

+6f=0,f(0) = 1,f′(0) =− 4 ,

(b)

d^2 f

dt^2

+2

df

dt

+5f=0,f(0) = 1,f′(0) = 0.

15.11 The quantitiesx(t),y(t) satisfy the simultaneous equations

̈x+2nx ̇+n^2 x=0,

̈y+2ny ̇+n^2 y=μx, ̇

wherex(0) =y(0) =y ̇(0) = 0 andx ̇(0) =λ. Show that

y(t)=^12 μλt^2

(

1 −^13 nt

)

exp(−nt).

15.12 Use Laplace transforms to solve, fort≥0, the differential equations

̈x+2x+y=cost,
̈y+2x+3y=2cost,
which describe a coupled system that starts from rest at the equilibrium position.
Show that the subsequent motion takes place along a straight line in thexy-plane.
Verify that the frequency at which the system is driven is equal to one of the
resonance frequencies of the system; explain why there isnoresonant behaviour
in the solution you have obtained.
15.13 Two unstable isotopesAandBand a stable isotopeChave the following decay
rates per atom present:A→B,3s−^1 ;A→C,1s−^1 ;B→C,2s−^1. Initially a
quantityx 0 ofAis present, but there are no atoms of the other two types. Using
Laplace transforms, find the amount ofCpresent at a later timet.
15.14 For a lightly damped (γ<ω 0 ) harmonic oscillator driven at its undamped
resonance frequencyω 0 , the displacementx(t)attimetsatisfies the equation

d^2 x

dt^2

+2γ

dx

dt

+ω^20 x=Fsinω 0 t.

Use Laplace transforms to find the displacement at a general time if the oscillator

starts from rest at its equilibrium position.

(a) Show that ultimately the oscillation has amplitudeF/(2ω 0 γ), with a phase

lag ofπ/2 relative to the driving force per unit massF.

(b) By differentiating the original equation, conclude that ifx(t) is expanded as

a power series intfor smallt, then the first non-vanishing term isFω 0 t^3 /6.

Confirm this conclusion by expanding your explicit solution.

15.15 The ‘golden mean’, which is saidto describe the most aesthetically pleasing
proportions for the sides of a rectangle (e.g. the ideal picture frame), is given
by the limiting value of the ratio of successive terms of the Fibonacci seriesun,
which is generated by
un+2=un+1+un,
withu 0 =0andu 1 = 1. Find an expression for the general term of the series and