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Chapter 7Chapter 7 LaplaceLaplaceTransformsTransforms

Laplace formulated Laplace's equation, and pioneered the

Pierre-Simon, marquis de LaplaceFrench mathematician & Astronomer

mathematical physics, a field that he took a leading role informing. The Laplacian differential operator, widely used in

mathematics, is also named after him. He restated and

developed the nebular hypothesis of the origin of the solar

system and was one of the first scientists to postulate theexistence of black holes and the notion of gravitational

collapse.

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7.1 Definition of Laplace Transform

{ } 0( ) ( ) ( )stt e f t dt F s = =L

is said to be a Laplace Transform offprovided that

Let fbe a function defined for t 0. Then the integral:

the integral converges

Lowercase letter is used to denote the function being

transformed, and the corresponding uppercase letter is to

denote its Laplace transform

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Table of Laplace Transform

{ }

{ }2

11 , 0

1, 0

ss

t ss

n

= >

= >

!

L

L

{ }

1 , a pos t ve nteger, 0

1,

n

at

t n ss

e s as a

+= >

= >

L

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Table of Laplace Transform

{ }

{ }

2 2

2 2

0

0

sin

cos

kkt s

s k

skt s

s k

= >+

= >+

L

L

{ }

{ }

2 2

2 2

sinh

cosh

kt s k s k

skt s k

s k

= >

= >

L

L

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Linearity of Laplace Transform

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

The Laplace Transform is a linear operator:

Example 1:

{ { {

2

1 5 1 5 1 5

1 15

t t t

ss

+ = + = +

= +

L L L L L

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Linearity of Laplace Transform

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

The Laplace Transform is a linear operator:

Example 2:

{ }

{ } { } { }

2 2

2

3 2

6 3 6 3

6 3 1

2 1 16 3

t t t t

t t

ss s

+ = +

= +

= +

L L L L

L L L

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Linearity of Laplace Transform

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

The Laplace Transform is a linear operator:

Exercise:

{ }

2 2

15 2 3 42

1

t

t

e t t

e

+

+

sin cos

( )

L

L

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7.2 Inverse Laplace Transform

{ }

{ }

1

1

2 2

1 11 1

1 1

s s

t ts s

= =

= =

L L

L L

{ }

11 1

1

1 1

n nn n

at at

n nt t

s s

e es a s a

+ +

= =

= =

L L

L L

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7.2 Inverse Laplace Transform

{ }

{ }

1

2 2 2 2

1

2 2 2 2

sin sin

cos cos

k kkt kt

s k s k

s skt kt

s k s k

= =

+ +

= =

+ +

L L

L L

{ }

{ }

12 2 2 2

1

2 2 2 2

sinh sinh

cosh cosh

k kkt kt s k s k

s skt kt

s k s k

= =

= =

L L

L L

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1

is a linear transform:

{ }1 ( ) ( )F s G s + = { }1 ( )F s + { }1 ( )G S

Example: 1

1 1 1 1

10

2 2s ss

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1

is a linear transform:

{ }1 ( ) ( )F s G s + = { }1 ( )F s + { }1 ( )G S

Example 2:

1 1 2

11

2 3s s

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Transform of Derivatives

= )0()()( 1)( fssFstf nnn )0(...)0( )1(2 nn ffs

eg.

{ } )(sYy =

12

ss =

{ } )0()0()(2 ysysYsy =

{ } )0()0()0()( 23 yysyssYsy =

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Use Laplace Transform to solve the IVP

1, (0) 0y y y = =

Example 1:

Example 2:

13

, ,y y e y y+ = = =