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Unit-3

Engineering Mathematics-II

Laplace Transform and its Applications

Introduction.

The Laplace transform is a powerful tool to solve differential equations. It transforms an Initial Value

Problem in Ordinary Differential Equation to algebraic equations.

Laplace Transform is an integral transform named after its inventor Pierre simon laplace.It Transforms a

Function of a real variable t to a function of a complex variable s.The transform has many application in

science and engineering. One important feature of the Laplace Transform is that it can transform analytic problems to algebraic. The concept of Laplace Transforms are applied in the area of science and technology such as Electric circuit analysis ,Communication engineering control engineering and nuclear physics etc.

Laplace Transform of a Function

f(t),represented by L[f(t)].

L[f(t)] =

0 )(dttfest

Formulae

[1]

0,1)1( ssL

[2] 1!)(n n s ntL [3] ,1)(aseLat [4] L( aseat 1) [5]

22][sinas

aatL [6]

22][cosas

satL [7]

22)(sinhas

aatL [8]

22)(coshas

satL

Theorem.1 If

`sLthensftFL1)1()()(

Proof: We know that

ss edteL dttFetFL st st st

1][1.1

0 0 0 f f

Hence proved

36
1 2 1 )}6(cos)1({2 1 }6cos12

1{3cos)(

)(cos497cos)( )(sin25

5)5(sin)(

.22/1

1)(12/3)(.

3cos)(7cos)(5sin)(

:1. 2 2 222
222

2/12/1

112/3
2/3 22/3
s s s tLL tLtLiv as satLs stLiii as aatLstLii Ansss s ntLstLisol tivtiiitiiti n n .12110].1[0 0.0 2,0 21,
10,0 )(2. 2 2 2 2 12 1 0 2 12 0

Ansssesses

e s te dtedttedte dttfetfLSol t tt t tfoftransformLaplacetheFindExamlpe ss stst ststst st f f Prop.-1 Laplace transform of derivative of order n ).0(....)0('')0()0()()}({3'21FFsFsFssfstfLnnnnn Prop.-2 Laplace transform of integral of the function f(t) )(1}{)()}({ 0 sfsdttfLthensftfLIf t

Prop.-3 Laplace Transform of

)(tftn (Multiplication by t) ......3,2,1)},({1)}(.{)()}({ nwheresfds dtftLthensftfLIfn nnn

Prop.-4 Laplace Transform of

t tf)( s dssft tfLthensftfLIf)(})({)()}({

Prop.-5 Linearity Property:

@>@>@)()()()(tgbLtfaLtbgtafL

Prop.-6 First shifting theorem

@@)()()()(asftfeLthensftfLIfat

Existence of Laplace transform:

The Laplace transform of a function

0),(ttf

can be found when 0 )(dttfest exists and this exists if the integral O 0 )(dttfest can actually be evaluated and its limit as .existsoO

Example.3

Find The Laplace transform of

tt2cos .)4( 4

41)2cos(

)}({1)}(.{,Pr )(4)2(cos. 22
2 2 1 2 Anss s s s ds dttLhavewe sfds dtftLtbytionmultiplicaofoperty sfs stLSinceSol n nnn

Example.4

Find The Laplace transform of

tet22 )(2

1)(2sfseLt

)}({1)}(.{,Prsfds dtftLtbytionmultiplicaofopertyn nnn 32
22
2 2 2

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