Principles of Mathematics in Operations Research

14.2 Laplace Transforms 193

Proposition 14.2.2 //J/:RHR satisfies

(i) y{t) = 0fort<0,

(ii) y(t) is piecewise continuous,

(Hi) y(t) = 0{eXot) for some x 0 6 M,

then y(t) has a Laplace transform.

Tables 14.1 and 14.2 contain Laplace transforms and its properties.

Table 14.1. A Brief Table for Laplace Transforms

Inverse Laplace Transform Valid s > xo

z/W v(s) xp

-J-,aeC Ka

s~a '

,I$r 0

b

(1)

(2)

(3)

(4).

(5)

(6)

(7)

(8)

tm,

1

eat

m = 1,2,.

tmeat, m = l,2

sin bt

cos bt

ect sin dt

ect cos dt

sa+6a^2

(s-c)^2 +d^2

( 3 -c)"^2 +d^2

Table 14.2. Properties of Laplace Transforms

(1)

(2)

(3)

(4)

(5)2/,

(6)

(7)

(8)

(9)

Inverse

y(t)

ay(t) + bz(t)

y'(t)

y{n)(t)

ft\ J 0, i < c where c > 0

e()~\l, t>c

se""»(s).<»>0

tmy(t), m=l,2,...

r^Ct)

/n J/(* - w)«(w) d"

Laplace Transform

7?(S)

otj(a) + b((s)

sri{s) - j/(0)

a"»j(a) - a—yo)

e-c8

s

7j(0S + &)

(-l)m7?(m)(s)

/s°° n(u) du

v(s)C(s)

Remark 14.2.3 If a = c+id is non-real, £{eat} = C{ect cos dt}+iC{edt sin dt}
then obtain Laplace transform using (2) in Table HA.

Remark 14.2.4 Proceed the following steps to solve an initial value problem: