Made By:-

S.Y. M-2

Shah Nisarg (130410119098)

Shah Kushal(130410119094)

Shah Maulin(130410119095)

Shah Meet(130410119096)

Shah Mirang(130410119097)

Laplace Transform And Its

Applications

Topics

Definition of Laplace Transform

Linearity of the Laplace Transform

Laplace Transform of some Elementary Functions

First Shifting Theorem

Inverse Laplace Transform

Laplace Transform of Derivatives & Integral

Differentiation & Integration of Laplace Transform

Evaluation of Integrals By Laplace Transform

Convolution Theorem

Application to Differential Equations

Laplace Transform of Periodic Functions

Unit Step Function

Second Shifting Theorem

Dirac Delta Function

Definition of Laplace Transform

Let f(t) be a given function of t defined for all

then the Laplace Transform ot f(t) denoted by L{f(t)}

or or F(s) or is defined as

provided the integral exists,where s is a parameter real

or complex.

0t

)(sf )(s

dttfessFsftfL st )()()()()}({0

Linearity of the Laplace Transform

If L{f(t)}= and then for any

constants a and b

)(sf )()]([ sgtgL

)]([)]([)]()([ tgbLtfaLtbgtafL

)]([)]([)}()({

)()(

)]()([)}()({

Definition-By :Proof

00

0

tgbLtfaLtbgtafL

dttgebdttfea

dttbgtafetbgtafL

stst

st

Laplace Transform of some Elementary

Functions

as if a-s

1

)(

e.)e(

Definition-By :Proof

a-s

1)L(e (2)

)0(,s

11.)1(

Definition-By :Proof

s

1L(1) (1)

0

)(

0

)(

0

atat

at

00

as

e

dtedteL

ss

edteL

tas

tasst

stst

|a|s, a-s

sat]L[cosh ly,(5)Similar

|a|s, a-s

a

11

2

1

)]()([2

1

2

eLat)L(sinh

definitionBy

2

eatcosh and

2

eatsinh have -We:Proof

a-s

aat]L[sinh (4)

-as, 1

]L[e 3)(

22

22

at

atat

22

at-

asas

eLeLe

ee

as

atatat

atat

0s, as

sat] L[cos and

as

aat]L[sin

get weparts,imaginary and real Equating

as

ai

as

s

as

ias

1

)L(e

1]e[]sin[cos

sincose

Formula] s[Euler' sincose that know -We:Proof

0s, as

sat] L[cos and

as

aat]L[sin (6)

2222

222222

at

iat

iat

ix

2222

asias

LatiatL

atiat

xix

n!1n 0,1,2...n n!

)(or

0,n -1n, 1

)(

1

ust ,.)-L(:Proof

n!or

1)()8(

1

0

1

1

0

1)1(

1

0

0

11

n

n

nx

n

n

nu

n

n

u

nstn

nn

n

StL

ndxxeS

ntL

duueS

s

du

s

ue

puttingdttet

SS

ntL

First Shifting Theorem

)(f]f(t)L[e ,

)(f]f(t)L[e

)(f)(f

ra-s where)(e

)(e

)(ef(t)]L[e

DefinitionBy Proof

)(f]f(t)L[ethen , (s)fL[f(t)] If

shifting-s theorem,shiftingFirst -Theorem

at-

at

0

rt-

0

a)t-(s-

0

st-at

at

asSimilarly

as

asr

dttf

dttf

dttfe

as

at

22)-(s

2-s)4cosL(e

2s

sL(cosh2t)

)2coshL(e (1)

43)(s

3s)4cosL(e

4s

sL(cos4t)

)4cosL(e (1)

:

22

2t

22

2t

22

3t-

22

3t-

t

t

t

t

Eg

Inverse Laplace Transform

)()}({L

by denoted is and (s)f of transformlaplace inverse

thecalled is f(t) then (s),fL[f(t)] If-Definition

1- tfsf

2

1

2

112

1

)2(

2

1

)1(

1

2

1C

than0s

2

1B

than-2s

-1A

than-1s

2)1)(sc(s1)(s)B(s2)(s)A(s1

)2()1())(2)(1(

1

2)(s)1)(s(s

1L )1(

21

1

1

tt eesss

L

If

If

If

s

C

s

B

s

A

sssL

Laplace Transform of Derivatives &

Integral

f(u)du(s)f1

L Also

(s)f1

f(u)duLthen (s),fL{f(t)} If

f(t) ofn integratio theof transformLaplace

(0)(0)....f fs-f(0)s-(s)fs(t)}L{f

f(0)-(s)fsf(0)-sL{f(t)}(t)} fL{

and 0f(t)elim provided exists, (t)} fL{ then

continous, piecewise is (t) f and 0 tallfor continous is f(t) If

f(t) of derivative theof transformLaplace

t

0

1-

t

0

1-n2-n1-nnn

st

t

s

s

22

2

22

3

22

2n

s

aat)L(sin

at)L(sin ss

a-

a-at)L(sin ssinat}L{-a

thisfrom a(0)f0,f(0) Also

sinat-a(t)f andat cos a(t)f sinat thenf(t)Let :Sol

atsin of transformlaplace DeriveExample

a

aa

)1(

1

)(1

cos

cosf(u) -Here:Sol

cos

2

0

0

ss

sfs

uduL

u

uduLEg

t

t

Differentiation & Integration of Laplace

Transform

0

n

nnn

ds (s)ft

f(t)Lthen

, transformLaplace has t

f(t) and (s) fL{f(t)} If

Transforms Laplace ofn Integratio

1,2,3,...n where, (s)]f[ds

d(-1)f(t)]L[t then (s) fL{f(t)} If

Tranform Laplace ofation Differenti

3

2

2

22at2

at2

)(

2

)(

1

1)1()e ( -:Sol

)e (:

as

asds

d

asds

dtL

tLExample

ss

s

s

s

ds

t

tLExample

s

11

11

1

s

22

cottan2

tantan

tan1

.t)L(sin -:Sol

sin

Evaluation of Integrals By Laplace

Transform

1)1()cos(

1)(cos

cos)cos(

cos)( 3

)()}({

cos -:Example

2

2

0

0

0

3

s

s

ds

dttL

s

stL

tdttettL

tttfs

dttfetfL

tdtte

st

st

t

25

2

100

8

)19(

19cos

cos)1(

1)cos(

)1(

2)1(1

2

0

3

0

22

2

22

22

tdtte

tdttes

sttL

s

ss

t

st

Convolution Theorem

g(t)*f(t)

g*fu)-g(t f(u)(s)}g (s)f{L

theng(t)(s)}g{L and f(t)(s)}f{L If

t

0

1-

-1-1

)1(e

e

.e

.)1(

1

)1(

1.

1

)1(

1

n theoremconvolutioby

)(1

1(s)g and )(

1(s)f have we:

)1(

1:

t

0

t

0

t

0

2

1

1

2

1

2

2

1

t

eue

dueu

dueuss

L

ssL

ssL

eLs

tLs

HereSol

ssLExample

tuu

t

u

t

ut

t

Application to Differential Equations

04L(y))yL(

sideboth on tranformLaplace Taking

.

.

(0)y-(0)ys-y(0)s-Y(s)s(t))yL(

(0)y-sy(0)-Y(s)s(t))yL(

y(0)-sY(s)(t))yL(

Y(s)L(y(t))

6(0)y 1y(0) 04yy :

23

2

eg

tts

s

2sin2

32cos

4s

6

4sY(s)

transformlaplace inverse Taking

4s

6Y(s)

06-s-4)Y(s)(s

04(Y(s))(0)y-sy(0)-Y(s)s

22

2

2

2

Laplace Transform of Periodic Functions

p

0

st 0)(sf(t)dt ee-1

1L{f(t)}

is p periodwith

f(t)function periodic continous piecewise a of transformlaplace The

0 tallfor f(t)p)f(t

if 0)p(

periodith function w periodic be tosaid is f(t)Afunction -Definition

ps-

2w

sπhcot

ws

w

e

e.

e1

e1.

ws

w

e1ws

w.

e1

1L[F(t)]

e1ws

w

wcoswt)ssinwt(ws

esinwtdteNow

tallfor f(t)w

πtf and

w

πt0for sinwt f(t)

0t|sinwt|f(t)

ofion rectificat wave-full theof transformlaplace theFind

22

2w

sπ

2w

sπ

w

sπ

w

sπ

22

w

sπ

22

w

sπ

w

sπ

22

2

w

π

0

w

π

0

22

stst

Unit Step Function

s

1L{u(t)}

0a if

es

1

s

e

(1)dte(0)dte

a)dt-u(tea)}-L{u(t

at1,

at0,a)-u(t

as-

a

st-

a

st-

a

0

st-

0

st-

Second Shifting Theorem

a))L(f(tea))-u(t L(f(t)-Corr.

L(f(t))e

(s)fea))-u(t a)-L(f(t

then(s)fL(f(t)) If

as-

as

as-

)(cos)2(

)2(cos)2()2(L

)()()(L

theroemshifting secondBy

(ii)L

33

1

}{.

}{)]2(L[e

2,ef(t)

)]2((i)L[e

22

1

22

21-

1-

22

21-

)3(2)62(

362

)2(323t-

3t-

-3t

ttu

ttus

sLtu

s

se

atuatfsfe

s

se

s

e

se

eLee

eLetu

a

tuExample

s

as

s

ss

ts

ts

Dirac Delta function

1))((

))((

0

1

0lim0

tL

eatL

tε , a

ε at , aε

at , - a)δ(t

as

ε

sin3tcos3t2ex(t)

sin3t2ecos3t2ex

inversion on

92)(s

6

92)(s

2)2(s

134ss

102sx

2(1)x13x(0)]-x4[s(0)]x-sx(0)x[s

have weTransform, Laplace Taking

0(0)x and 2x(0)0,t(t),213xx4x

0(0)x and 2x(0)0,at t here w

(t)213xx4xequation the-Solve:Example

2t

2t-2t-

222

2