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6. Laplace Transform Properties

Dr. Samir Al-Amer

Reading Assignment :

CISE302: Linear Control Systems

Term 081

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Learning Objective

To be able to use Laplace transform to

solve linear constant coefficient ordinary

differential equations

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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

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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

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Laplace Transform ofDerivative

)0()0()0()()(

)0()0()()(

)0()()(

23

3

3

2

2

2

ffsfssFsdt

tfdL

fsfsFsdt

tfdL

fssFdt

tdfL

!

!

!

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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

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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)

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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

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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

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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

!

!

!!

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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

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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

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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