Higher Engineering Mathematics

PROPERTIES OF LAPLACE TRANSFORMS 635

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assuming e−stf′(t)→0ast→∞, andf′(0) is the
value off′(t)att=0. Hence
{f′′(t)}=−f′(0)+s[s(f(t))−f(0)], from equation (3),

i.e.

L{f′′(t)}

=s^2 L{f(t)}−sf(0)−f′(0)

or L

{

d^2 y

dx^2

}

=s^2 L{y}−sy(0)−y′(0)

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(4)

wherey′(0) is the value of

dy

dx

atx=0.

Equations (3) and (4) are important and are
used in the solution of differential equations (see
Chapter 67) and simultaneous differential equations
(Chapter 68).

Problem 5. Use the Laplace transform of the

first derivative to derive:

(a)L{k}=

k

s

(b)L{ 2 t}=

2

s^2

(c)L{e−at}=

1

s+a

From equation (3),L{f′(t)}=sL{f(t)}−f(0).

(a) Letf(t)=k, thenf′(t)=0 andf(0)=k.

Substituting into equation (3) gives:

L{ 0 }=sL{k}−k

i.e. k=sL{k}

Hence L{k}=

k

s

(b) Letf(t)= 2 tthenf′(t)=2 andf(0)=0.

Substituting into equation (3) gives:

L{ 2 }=sL{ 2 t}− 0

i.e.

2

s

=sL{ 2 t}

Hence L{ 2 t}=

2

s^2

(c) Letf(t)=e−atthenf′(t)=−ae−atandf(0)=1.

Substituting into equation (3) gives:

L{−ae−at}=sL{e−at}− 1

−aL{e−at}=sL{e−at}− 1

1 =sL{e−at}+aL{e−at}

1 =(s+a)L{e−at}

HenceL{e−at}=

1

s+a

Problem 6. Use the Laplace transform of the

second derivative to derive

L{cosat}=

s

s^2 +a^2

From equation (4),

L{f′′(t)}=s^2 L{f(t)}−sf(0)−f′(0)

Letf(t)=cosat, thenf′(t)=−asinatand

f′′(t)=−a^2 cosat,f(0)=1 andf′(0)= 0

Substituting into equation (4) gives:

L{−a^2 cosat}=s^2 {cosat}−s(1)− 0

i.e. −a^2 L{cosat}=s^2 L{cosat}−s

Hence s=(s^2 +a^2 )L{cosat}

from which, L{cosat}=

s

s^2 +a^2

Now try the following exercise.

Exercise 233 Further problems on the

Laplace transforms of derivatives

1. Derive the Laplace transform of the first
   derivative from the definition of a Laplace
   transform. Hence derive the transform

L{ 1 }=

1

s

2. Use the Laplace transform of the first deriva-
   tive to derive the transforms:

(a)L{eat}=

1

s−a

(b)L{ 3 t^2 }=

6

s^3

3. Derive the Laplace transform of the second
   derivative from the definition of a Laplace
   transform. Hence derive the transform

L{sinat}=

a

s^2 +a^2