Ordinary Differential Equations[FCM 1023]

LAPLACE TRANSFORM

Chapter 5

Overview

Chapter 5: LAPLACE TRANSFORM

5.1. Definition and Basic Properties

5.2. Inverse Laplace Transform

5.3. Method of Solution

5.4. Operational Properties

Learning OutcomeAt the end of this section, you should be able to:

Use translation theorems

5.4 Operational Properties

If and a is any real number, then

5.4 Operational Properties

First Translation Theorem

)()}({ sFtfL

)()}({ asFtfeL at

assat tfLtfeL )}({)}({

.

}{ 35 teL t

5335 }{}{ ss

t tLteL

Example 15.4 Operational Properties

54!3

sss

4)5(6

s

Evaluate

assat tfLtfeL )}({)}({

.

Example 25.4 Operational Properties

}4cos{ 2 teL t

22 }4{cos}4cos{

sst tLteL

222 4

sss

s

16)2(

22

ss

Evaluate

assat tfLtfeL )}({)}({

.

)()}({ asFtfeL at

)(})({)}({ 11 tfesFLasFL atass

5.4 Operational Properties

Inverse of First Translation Theorem

.

21

)3(52

ssL

22 )3(11

32

)3(52

ssss

211

21

)3(111

312

)3(52

sL

sL

ssL

32

13

)(1112

ss

t

sLe

tt tee 33 112

Example 15.4 Operational Properties

)(})({1 tfesFL atass

Evaluate

.

11621

sssL

Example 25.4 Operational Properties

23116 21

21

ssL

sssL

233

233

221

sssL

2333

21

ssL

Evaluate

.

Example 25.4 Operational Properties

233

233

221

sssL

221

221

23

1323

3

sL

s

sL

322

1

322

1

2

223

2 ssss sL

s

sL

tete tt 2sin232cos 33 )(})({1 tfesFL at

ass

Exercise5.4 Operational Properties

821

11

231

sssLEvaluate

tete tt 3sinh31

21 2 Ans:

The unit step function, is defined as

5.4 Operational Properties

Unit Step Function or Heaviside Function

)( atU

.,10,0

)(at

atatU

U

t

1

a

If and then

5.4 Operational Properties

Second Translation Theorem

)}({)( tfLsF ,0a

)()}()({ sFeatUatfL as

seatUL

as

)}({

.

22 3 tUtL

Example 15.4 Operational Properties

323 22 tLetUtL s

42 !3

se s

4

26se s

Evaluate )()}()({ sFeatUatfL as

32)2( ttf 3)( ttf 2a

.

3232 tUtUtf

Example 25.4 Operational Properties

Find the Laplace Transform

3232 tULtULLtfLs

eatULas

)}({

se

se

s

ss 32

32

.

132 tUttg

Example 35.4 Operational Properties

Find the Laplace Transform

132 tUtLtgL )()}()({ sFeatUatfL as

12 tLe s

12 LtLe s

sse s 12

2

The unit step function can be used to write piecewise defined function in a compact form.

For instance:

The compact is:

5.4 Operational Properties

.,0,

atthattg

tf

atUthatUtgtgtf

For instance:

The compact is:

5.4 Operational Properties

bt

btatgat

tf,0,0,0

btUatUtgtf

Find

3,230,

)( t

t ttf)}({ tfL where

Example 45.4 Operational Properties

atth

attgtf

,0,

atUthatUtgtgtf

Solution:Step 1: Change to compact form

ttg )(

atUthatUtgtgtf

5.4 Operational Properties

3,230,

)( t

t ttf

2)( th3a

323 tUttUttf

5.4 Operational Properties

)}3({2)}3({}{)}({ tULttULtLtfL

settUL

s

s3

2 2)}3({1

s

eatULas

)}({

)}3({ ttUL

Step 2: Compute the Laplace transform

323 tUttUttf

)()}()({ sFeatUatfL as

5.4 Operational Properties

ttf )3( 3)( ttf

)()}3({ 3 sFettUL s

sstLtfLsF 31}3{)}({)( 2

)}3({ ttUL

3a

)()}()({ sFeatUatfL as

ssettUL s 31)}3({ 23

se

se ss 3

2

31

s

ettULs

tfLs3

2 2)}3({1

se

se

se

s

sss 33

2

3

2231

5.4 Operational Properties

If then )}({)( 1 sFLtf

).()()}({1 atUatfsFeL as

5.4 Operational Properties

Inverse of Second Translation Theorem

Example 15.4 Operational Properties

Evaluate }41{ 21 se

sL

}41{ 21 se

sL

).()()}({1 atUatfsFeL as

2a41)(

s

sF

tfes

L t

41 }41{

5.4 Operational Properties

)2(}41{ )2(421

tUees

L ts

Therefore,

tetf 4)( 24)2( tetf

)()()}({1 atUatfsFeL as

tfes

L t

41 }41{