Higher Engineering Mathematics

Laplace transforms

65

Properties of Laplace transforms

65.1 The Laplace transform of eatf(t)

From Chapter 64, the definition of the Laplace

transform off(t) is:

L{f(t)}=

∫∞

0

e−stf(t)dt (1)

Thus L{eatf(t)}=

∫∞

0

e−st(eatf(t)) dt

=

∫∞

0

e−(s−a)f(t)dt (2)

(whereais a real constant)

Hence the substitution of (s−a) forsin the trans-

form shown in equation (1) corresponds to the

multiplication of the original functionf(t)byeat.

This is known as a shift theorem.

65.2 Laplace transforms of the form

eatf(t)

From equation (2), Laplace transforms of the form

eatf(t) may be deduced. For example:

(i)L{eattn}

SinceL{tn}=

n!

sn+^1

from (viii) of Table 64.1,

page 628.

thenL{eattn}=

n!

(s−a)n+^1

from equation (2)

above (provideds>a).

(ii)L{eatsinωt}

Since L{sinωt}=

ω

s^2 +ω^2

from (iv) of

Table 64.1, page 628.

thenL{eatsinωt}=

ω

(s−a)^2 +ω^2

from equa-

tion (2) (provideds>a).

(iii)L{eatcoshωt}

SinceL{coshωt}=

s

s^2 −ω^2

from (ix) of

Table 64.1, page 628.

thenL{eatcoshωt}=

s−a

(s−a)^2 −ω^2

from equa-

tion (2) (provideds>a).

A summary of Laplace transforms of the form eatf(t)

is shown in Table 65.1.

Table 65.1 Laplace transforms of the form eatf(t)

Function eatf(t) Laplace transform

(ais a real constant) L{eatf(t)}

(i) eattn

n!

(s−a)n+^1

(ii) eatsinωt

ω

(s−a)^2 +ω^2

(iii) eatcosωt

s−a

(s−a)^2 +ω^2

(iv) eatsinhωt

ω

(s−a)^2 −ω^2

(v) eatcoshωt

s−a

(s−a)^2 −ω^2

Problem 1. Determine (a)L{ 2 t^4 e^3 t}

(b)L{4e^3 tcos 5t}.

(a) From (i) of Table 65.1,

L{ 2 t^4 e^3 t}= 2 L{t^4 e^3 t}= 2

(

4!

(s−3)^4 +^1

)

=

2(4)(3)(2)

(s−3)^5

=

48

(s−3)^5

(b) From (iii) of Table 65.1,

L{4e^3 tcos 5t}= 4 L{e^3 tcos 5t}

= 4

(

s− 3

(s−3)^2 + 52

)