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SECOND ORDER LINEARORDINARY DIFFERENTIAL

EQUATIONS WITH

CONSTANT COEFFICIENT

LAPLACE

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Lets see

Its easier to get the answer

Table!

http://localhost/var/www/apps/conversion/tmp/scratch_4/TABLE.dochttp://localhost/var/www/apps/conversion/tmp/scratch_4/TABLE.doc

5/28/2018 Laplace Transformation

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FSTPi, UTHM

Theorem 1 : Linearity

Characteristics of Laplace Transform

where and are costants.

)}()({ 21 tftf )}({ 1 tf )}({ 2 tfL L L

Theorem 2 : First Shift

).()}({ asFtfe at

If {f(t)}= F(s), and a is a constant,then

L

L

Example

Example

PROPERTIES OF INVERSE LAPLACE TRANSFORM

http://localhost/var/www/apps/conversion/tmp/scratch_4/CHARACTERISTICS%20linearity.dochttp://localhost/var/www/apps/conversion/tmp/scratch_4/CHARACTERISTICS%201st%20shift.dochttp://localhost/var/www/apps/conversion/tmp/scratch_4/CHARACTERISTICS%201st%20shift.dochttp://localhost/var/www/apps/conversion/tmp/scratch_4/CHARACTERISTICS%20linearity.doc

5/28/2018 Laplace Transformation

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FSTPi, UTHM

nt

n

nn

ds

d)1(

n

nn

ds

d)1(

Theorem 3 : Multiply with

If L{f(t)} = F(s), then for n = 1, 2, 3, ..

L{ f(t)} = L{f(t)}

F(s)=

Example

Try this!

http://localhost/var/www/apps/conversion/tmp/scratch_4/CHARACTERISTICS%20multyply.dochttp://localhost/var/www/apps/conversion/tmp/scratch_4/CHARACTERISTICS%20Ex.dochttp://localhost/var/www/apps/conversion/tmp/scratch_4/CHARACTERISTICS%20Ex.dochttp://localhost/var/www/apps/conversion/tmp/scratch_4/CHARACTERISTICS%20multyply.doc

5/28/2018 Laplace Transformation

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FSTPi, UTHM

)()}({1 tfsF

1

1

INVERSE LAPLACE TRANSFORMATIONDefinition : Inverse Laplace Transform

If L{f(t)} = F(s), so L

L

is known as the operation ofinverse Laplace transform.

Note : L

1L

Example

http://localhost/var/www/apps/conversion/tmp/scratch_4/INVERSE%20LAPLACE.dochttp://localhost/var/www/apps/conversion/tmp/scratch_4/INVERSE%20LAPLACE.doc

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How to apply these techniques in findingInverse Laplace transforms?

Step 1 Determine either the given function need thesetechniques or not.

Step 2 Apply techniques of partial fraction or completing thesquare.

Step 3 Find inverse Laplace transformation for the newFunction.

5/28/2018 Laplace Transformation

7/13FSTPi, UTHM

METHOD OF PARTIAL FRACTION ANDCOMPLETING THE SQUARE

Sometimes, we need to apply technique of partial fractionand completing the squarebefore we can proceed withinverse Laplace transformation.

Partial fraction 1. Rational function --> DegreeD (s) >N (s)

if not, doing long division first.2. FactorizeD(s)if possible.3. Refer table of partial fraction.

( )

( )

N s

D s

Completing the square Example:2 2

2 2

6 8 ( 3) 1

2 3 ( 1) 2

s s s

s s s

5/28/2018 Laplace Transformation

8/13FSTPi, UTHM

Partial Fraction

The denominator Partial fraction

ax b Aax b

2( )ax b 1 2 2( )A Aax b ax b ( )rax b 1 2 2( ) ( )r rA A Aax b ax b ax b

2ax bx c 2Ax Bax bx c 2 2

( )ax bx c 1 1 2 2

2 2 2( )

A x B A x B

ax bx c ax bx c

2( )rax bx c 1 1 2 22 2 2 2

( ) ( )

r r

r

A x B A x B A x B

ax bx c ax bx c ax bx c

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1

Determine L

Example :

Do you think the expression above easier toget the answer from table?

See more detail .

6

132 ss

s

http://localhost/var/www/apps/conversion/tmp/scratch_4/PARTIAL%20FRACTION.dochttp://localhost/var/www/apps/conversion/tmp/scratch_4/PARTIAL%20FRACTION.doc

5/28/2018 Laplace Transformation

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Application Laplace Transform to Solve

Nonhomogenous Deferential Equation

)()}({ sYty L

)0()()}({ yssYty L

)0()0()()}({ 2 yyssYsty L

5/28/2018 Laplace Transformation

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Steps of solving :

Take Laplace transform of both sides of the equation. Obtain the algebraic equation for Y(s).

)(xy )}({1 sY

)(ty )}({1 sY

Solve for Y(s).

Take the inverse Laplace transform to get :

L

or

L

5/28/2018 Laplace Transformation

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Solve

,42 ydt

dy given y(0) = 1.1.

2. teydt

dy

dt

yd 3

2

2

223 giveny(0) = 5 andy(0) = 7.

See more detail .

See more detail .

http://localhost/var/www/apps/conversion/tmp/scratch_4/Application%202nd%20order.dochttp://localhost/var/www/apps/conversion/tmp/scratch_4/Application%201st%20order.dochttp://localhost/var/www/apps/conversion/tmp/scratch_4/Application%201st%20order.dochttp://localhost/var/www/apps/conversion/tmp/scratch_4/Application%202nd%20order.doc

5/28/2018 Laplace Transformation

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LETS TRY THIS!

Pierre-Simon, marquis de Laplace(1745-1827)

http://localhost/var/www/apps/conversion/tmp/scratch_4/Exercise%20Application.dochttp://upload.wikimedia.org/wikipedia/commons/1/16/Pierre-Simon_Laplace_signature.svghttp://upload.wikimedia.org/wikipedia/commons/b/bf/Pierre-Simon%2C_marquis_de_Laplace_%281745-1827%29_-_Gu%C3%A9rin.jpghttp://localhost/var/www/apps/conversion/tmp/scratch_4/Exercise%20Application.doc