Mathematical Methods for Physics and Engineering : A Comprehensive Guide

13.4 EXERCISES

(c) L[sinhatcosbt]=a(s^2 −a^2 +b^2 )[(s−a)^2 +b^2 ]−^1 [(s+a)^2 +b^2 ]−^1.

13.24 Find the solution (the so-calledimpulse responseorGreen’s function)ofthe
equation

T

dx

dt

+x=δ(t)

by proceeding as follows.

(a) Show by substitution that

x(t)=A(1−e−t/T)H(t)

is a solution, for whichx(0) = 0, of

T

dx

dt

+x=AH(t), (∗)

whereH(t) is the Heaviside step function.

(b) Construct the solution when the RHS of (∗) is replaced byAH(t−τ), with

dx/dt=x=0fort<τ, and hence find the solution when the RHS is a

rectangular pulse of durationτ.

(c) By settingA=1/τand taking the limit asτ→0, show that the impulse

response isx(t)=T−^1 e−t/T.

(d) Obtain the same result much more directly by taking the Laplace transform

of each term in the original equation, solving the resulting algebraic equation

and then using the entries in table 13.1.

13.25 This exercise is concerned with the limiting behaviour of Laplace transforms.

(a) Iff(t)=A+g(t), whereAis a constant and the indefinite integral ofg(t)is

bounded as its upper limit tends to∞, show that

lim

s→ 0

s ̄f(s)=A.

(b) Fort>0, the functiony(t) obeys the differential equation

d^2 y

dt^2

+a

dy

dt

+by=ccos^2 ωt,

wherea,bandcare positive constants. Find ̄y(s)andshowthats ̄y(s)→c/ 2 b

ass→0. Interpret the result in thet-domain.

13.26 By writingf(x) as an integral involving theδ-functionδ(ξ−x) and taking the
Laplace transforms of both sides, show that the transform of the solution of the
equation
d^4 y
dx^4

−y=f(x)

for whichyand its first three derivatives vanish atx= 0 can be written as

̄y(s)=

∫∞

0

f(ξ)

e−sξ

s^4 − 1

dξ.

Use the properties of Laplace transforms and the entries in table 13.1 to show

that

y(x)=

1

2

∫x

0

f(ξ)[sinh(x−ξ)−sin(x−ξ)]dξ.