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8/8/2019 CISE302 081 Lecture6 Solving Differential Equations 0
1/15
CISE302_L6 Dr. Samir Al-Amer 2008 1
6. Laplace Transform Properties
Dr. Samir Al-Amer
Reading Assignment :
CISE302: Linear Control Systems
Term 081
8/8/2019 CISE302 081 Lecture6 Solving Differential Equations 0
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CISE302_L6 Dr. Samir Al-Amer 2008 2
Learning Objective
To be able to use Laplace transform to
solve linear constant coefficient ordinary
differential equations
8/8/2019 CISE302 081 Lecture6 Solving Differential Equations 0
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CISE302_L6 Dr. Samir Al-Amer 2008 3
Use of Laplace Transform in solving ODE
Differential Equation
Laplace
Transform
Algebraic Equation
Solution of the
Algebraic Equation
InverseLaplace
transform
Solution of the
Differential Equation
2
1)()(
0)(21)(1)0(,0)(2)(
2
!!
!!!
ssXetx
sXssXxtxtx
t
8/8/2019 CISE302 081 Lecture6 Solving Differential Equations 0
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CISE302_L6 Dr. Samir Al-Amer 2008 4
Solution Procedure
1. Apply Laplace transform to thedifferential equation to obtain analgebraic equation
2. Solve the algebraic equation for theunknown function
3. Use Partial fraction expansion to expressthe unknown function as the sum ofsimple terms
4. Use inverse Laplace transform to obtainthe solution of the original problem
8/8/2019 CISE302 081 Lecture6 Solving Differential Equations 0
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CISE302_L6 Dr. Samir Al-Amer 2008 5
Laplace Transform ofDerivative
)0()0()0()()(
)0()0()()(
)0()()(
23
3
3
2
2
2
ffsfssFsdt
tfdL
fsfsFsdt
tfdL
fssFdt
tdfL
!
!
!
8/8/2019 CISE302 081 Lecture6 Solving Differential Equations 0
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0)0(,2)(4)(
ODEthesolvetotrasformLaplaceApply
!! xtxtx
Solving ODEExample 1
_ a
_ a
_ a
ssxss
sL
stxL
xsstxL
2)(4)0()(
2
2
)()(
)0()()(
!
!
!
!Step 1: Apply Laplace transform to the ODE
8/8/2019 CISE302 081 Lecture6 Solving Differential Equations 0
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CISE302_L6 Dr. Samir Al-Amer 2008 7
0)0(,2)(4)(
ODEthesolvetotrasformLaplaceApply
!! xtxtx
Solving ODEExample 1
)4(
2)(
2)()4(
2)(4)(
!
!
!
sssX
s
sXs
ssXssX
Step 2: Solve for the unknown function X(s)
8/8/2019 CISE302 081 Lecture6 Solving Differential Equations 0
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CISE302_L6 Dr. Samir Al-Amer 2008 8
0)0(,2)(4)(
ODEthesolvetotrasformLaplaceApply
!! xtxtx
Solving ODEExample 1
4
5.05.0)(
5.0)4(
2)4(
5.0)4(
2)(
4)4(
2)(
4
0
!
!
!
!
!
!
!
!
!
sss
sssB
sssA
s
B
s
A
sss
s
s
Step 3: Partial Fraction Expansion
8/8/2019 CISE302 081 Lecture6 Solving Differential Equations 0
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0)0(,2)(4)(
ODEthesolvetotrasformLaplaceApply
!! xtxtx
Solving ODEExample 1
tetx
sss
45.05.0)(
4
5.05.0)(
!
!
Step 4: Inverse Laplace Transform
8/8/2019 CISE302 081 Lecture6 Solving Differential Equations 0
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5)0(,4)0(,1)(2)(3)(
ODEthesolvetotrasformLaplaceApply
!!! xxtxtxtx
Solving ODEExample 2
_ a_ a
_ a_ a
sL
stxL
xsstxL
xsxsstxL
11
)()(
)0()()(
)0()0()()( 2
!
!
!!
8/8/2019 CISE302 081 Lecture6 Solving Differential Equations 0
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Solving ODEExample 2 Step 1: Apply Laplace Transform
_ a _ a
? A
? A ssXssXssXs
ssXxssXxsxsXs
tutxtxtx
STEP
1
)(2]4)([354)(
1)(2)]0()([3)0()0()(
)()(2)(3)(
:
2
2
!
!
!
5)0(,4)0(,1)(2)(3)(
ODEthesolvetotrasformLaplaceApply
!!! xxtxtxtx
8/8/2019 CISE302 081 Lecture6 Solving Differential Equations 0
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Solving ODEExample 2 Step 2: Solve for X(s)
? A
? A
)23(
1174
23
1174
)(
11254)(23
1)(2]4)([354)(
:2
2
2
2
2
2
!
!
!
!
sss
ss
ss
ss
sX
sssXss
ssXssXssXs
STEP
5)0(,4)0(,1)(2)(3)(
ODEthesolvetotrasformLaplaceApply
!!! xxtxtxtx
8/8/2019 CISE302 081 Lecture6 Solving Differential Equations 0
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Solving ODEExample 2 Step 3: Partial Fraction Expansion
5.9)23(
1174)2(
,14)23(1174)1(
,5.0)23(
1174
21)23(1174)(
:3
2
2
2
1
2
2
0
2
2
2
2
!
!
!!
!
!
! !
!
!
!
s
s
s
sss
sssC
ssssssB
sss
sssA
s
C
s
B
s
A
sss
sss
S
5)0(,4)0(,1)(2)(3)(
ODEthesolvetotrasformLaplaceApply
!!! xxtxtxtx
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Solving ODEExample 2 Step 4: Inverse Laplace transform
05.9145.0)(
2
5.9
1
145.0)(
:4
2
u!
!
tforeetx
ssssX
STEP
tt
5)0(,4)0(,1)(2)(3)(
ODEthesolvetotrasformLaplaceApply
!!! xxtxtxtx
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Summary
1. Apply Laplace transform to thedifferential equation
2. Solve for the unknown function
3. Use Partial fraction expansion toexpress the unknown function as thesum of simple terms
4. Use inverse Laplace transform toobtain the solution of the originalproblem