ode_chapter 05-04 [jan 2014]
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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{
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