Mathematical Methods for Physics and Engineering : A Comprehensive Guide

HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS

15.29 The equation of motion for a driven damped harmonic oscillator can be written

̈x+2x ̇+(1+κ^2 )x=f(t),
withκ=0.Ifitstartsfromrestwithx(0) = 0 andx ̇(0) = 0, find the corresponding
Green’s functionG(t, τ) and verify that it can be written as a function oft−τ
only. Find the explicit solution when the driving force is the unit step function,
i.e.f(t)=H(t). Confirm your solution by taking the Laplace transforms of both
it and the original equation.
15.30 Show that the Green’s function for the equation

d^2 y

dx^2

+

y

4

=f(x),

subject to the boundary conditionsy(0) =y(π) = 0, is given by

G(x, z)=

{

−2cos^12 xsin^12 z 0 ≤z≤x,

−2sin^12 xcos^12 zx≤z≤π.

15.31 Find the Green’s functionx=G(t, t 0 ) that solves

d^2 x

dt^2

+α

dx

dt

=δ(t−t 0 )

under the initial conditionsx=dx/dt=0att=0.Hencesolve

d^2 x

dt^2

+α

dx

dt

=f(t),

wheref(t)=0fort<0.
Evaluate your answer explicitly forf(t)=Ae−at(t>0).
15.32 Consider the equation

d^2 y

dx^2

+f(y)=0,

wheref(y) can be any function.

(a) By multiplying through bydy/dx, obtain the general solution relatingxand

y.

(b) A massm, initially at rest at the pointx= 0, is accelerated by a force

f(x)=A(x 0 −x)

[

1+2ln

(

1 −

x

x 0

)]

.

Its equation of motion ismd^2 x/dt^2 =f(x). Findxas a function of time, and

show that ultimately the particle has travelled a distancex 0.

15.33 Solve

2 y

d^3 y

dx^3

+2

(

y+3

dy

dx

)

d^2 y

dx^2

+2

(

dy

dx

) 2

=sinx.

15.34 Find the general solution of the equation

x

d^3 y

dx^3

+2

d^2 y

dx^2

=Ax.

15.35 Express the equation

d^2 y

dx^2

+4x

dy

dx

+(4x^2 +6)y=e−x

2

sin 2x

in canonical form and hence find its general solution.