Trang 1

MC LC

MC LC____________________________________________________________________1I. NH NGHA PHP BIN I Laplace: _____________________________________2

A. HM GC:__________________________________________________________________ 2

B. PHP BIN I Laplace ______________________________________________________ 2

C. MT S TNH CHT CA BIN I Laplace: __________________________________ 3V d: _________________________________________________________________________________ 3

D. PHP BIN I Laplace NGC: _____________________________________________ 4nh ngha:_____________________________________________________________________________ 4

II. NG DNG Laplace GII PHNG TRNH VI PHN THNG:______________5A. PHNG PHP CHUNG: _____________________________________________________ 5B. CC V D: _________________________________________________________________ 6

V d 1: ________________________________________________________________________________ 6V d 2: ________________________________________________________________________________ 7V d 3: ________________________________________________________________________________ 8

III. NG DNG Laplace GII PHNG TRNH VI PHN C V PHI L HM BC THANG: _____________________________________________________________________9

1) nh ngha: __________________________________________________________________ 9

2) Bin i Laplace: ____________________________________________________________ 10V d: ________________________________________________________________________________ 11

3) Bin i Laplace ngc: ______________________________________________________ 12V d 1: _______________________________________________________________________________ 12V d 2: _______________________________________________________________________________ 14

IV. NG DNG Laplace GII H PHNG TRNH VI PHN H S HNG ______15A. PHNG PHP CHUNG: ____________________________________________________ 15

B. CC V D: ________________________________________________________________ 15V d 1: _______________________________________________________________________________ 15V d 2: _______________________________________________________________________________ 16V d 3: _______________________________________________________________________________ 17

V. KT LUN: _______________________________________________________________20

Trang 2

DNG BIN I LAPLACE GII PHNG TRNH-

H PHNG TRNH VI PHN H S HNG

Trong phn tiu lun ny chng ta dng php bin i Laplace lm mt k thut

khc gii phng trnh-h phng trnh vi phn tuyn tnh h s hng. N cng l mt

k thut c bit gii phhng trnh vi phn c v phi l hm bc thang Heaviside1.

Nhng hm ny thng xut hin trong chc v trong mch in t.

tng ca phng php ny l: Bin i phng trnh vi phn thnh phng trnh

i s, gii phng trnh i s va bin i , t nghim ca phng trnh i s va

tmc ta dng bin i ngc Laplace cho ra nghim phng trnh vi phn cn tm.

I. NH NGHA PHP BINI Laplace:

A. HM GC:

Ta gi hm phc ty )(tf l hm gc tho mn 3iu kin sau:

1) Hu hn im ,0,ba2) Tng khng qu nhanh 0

S0 ,.)(0,0 0 tteMtfSM

t , S0c gi l m

tng ca hm )(tf

3) )(tf =0 khi t

Trang 3

C. MT STNH CHT CA BINI Laplace:1) Cho 2 Laplace )(tf , g(t); )(tf = F(p); g(t) = G(p)

)(tf +g(t) = F(p) + G(p)

2) )(tf , k l hng s

k. )(tf = k.F(p)

3) o hm gc:

)0(...)0(')0()()(

)0(')0()()(''

)()0()0(),0()()('

)()(

)1(21)(

2

0lim

nnnnn

t

ffpfppFptf

fpfpFptf

tffffppFtf

pFtf

Chng Minh:Ta c:

)()0(0

)()(')('000

)(ppFf

dttfepptdttfetf ptpt tfe

)0(')0()(

)0(')0()()(''2 fpfpFp

ffppFptf

Tng tcho )()( tf n

4) Tnh tin nh

tconsaapFtfeL

pFtfLat tan)()(

)()(

Chng minh:

0

)(

0

)()()(.)( apFdttfedttfeetfeL tapatptat

V d:

Bin i Laplace:

a)pp

edtept

pt 1)(

1.100

Trang 4

b)papa

edtedteeetpa

tpaatptat

1)( 00)(

)(

0

c)2

02

000

1pp

edtp

ep

tetdtetptptpt

pt

D. PHP BIN I Laplace NGC:

nh ngha:Cho nh )(pF tm gc )(tf

)()(1 tfpFL BNG I CHIU CC BIN I Laplace THNG DNG

)(tf )( pF

1 0,1 pp

t 0,1

2 ppnt np

pnn ,0,!

1 l stnhin

21

tp

, p>0

21

t 2/32 p

)21(n

tpp

nnn

2

)12...(5.3.1 ,p>0, n l stnhin

ate apap

,1

ate apap

,1

atte apap

,)(1

2

Trang 5

atte apap

,)(1

2

atnet napap

nn ,,)(

!1 l stnhin

atnet napap

nn ,,)(

!1 l stnhin

atcos 0,22 papp

atsin 0,22 papa

att cos0,

)( 22222

papap

att sin 0,)(

2222 pap

ap

bte at sin apbap

b ,)( 22

bte at cos apbap

ap

,)( 22

atcosh apap

p ,22

atsinhap

apa ,22

II. NG DNG Laplace GII PHNG TRNH VI PHN THNG:

A. PHNG PHP CHUNG:Cho Phng trnh vi phn tuyn tnh h s hng c dng:

)()()('...)()( 01)(

1)( tftyatyatyatya nn

nn

Trong Raaa n ...,,, 10

1)1(

10 ,...,)0(',)0( nn bybyby l nhng iu kin u

Php bin i trc tip Laplace khng cho nghim tng qut. Cc bc gii l:

Trang 6

1) nh gi Laplace da vo 2 mt ca phng trnh.

2) S dng bng bin i Laplace cbn.

)0(...)0(.)(.))(( )1(1)( nnnn ffppFptfL

3) Sau qu trnh bin i i s ta c:Y(p) = L(y(t))

4) Lm php bin i ngc Laplace L-1, tm y(t).

B. CC V D:

V d 1:

Tm nghim phng trnh vi phn

1)0(',1)0(2'3'' 3

yyeyyy t

Gii

Sdng tnh chto hm gc v bini Laplace:

)3)(2)(1(1

11

)(

31

2)()2)(1(

312)()23(

31)(23)(31)(

31

)(2)0()(3)0(')0()(

2

2

2

pppppY

pppYpp

pppYpp

ppYppYppYp

ppYyppYypypYp

Dng phng phpi sphn tch:

)3)(2)(1()236()345()(

)3)(2)(1()2)(1()3)(1()3)(2(

)3()2()1()3)(2)(1(1

2

pppCBApCBApCBA

pppppCppBppA

pC

pB

pA

ppp

Cn bng hs2 v: cho21

,1,21 CBA

Trang 7

)3(2/1

)2(1

)1(2/1

)3)(2)(1(1

pppppp

31

21

21

11

23

)( ppppY

Sdng bini ngc Laplace

Vy nghim phng trnh l:

ttt eeety 3221

23)(

V d 2:

Tm nghim ca phng trnh vi phn:

0)0('',0)0(',0)0(''''

yyyeyy t

Gii

S dng tnh cht o hm gc v bin i Laplace:

))(1(1

)(

11

)(.)(

11

)0()(.)0('')0(.)0(''.)(.

3

3

23

ppppY

ppYpp

pypYpyypyppYp

Dng phng phpi s phn tch vphi

)1)(1()()()(

)1)(1()1()()1()1)(1(

11))(1(1

2

23

2

22

23

pppApDBApDCApCBA

pppppDCppBpppA

pDCp

pB

pA

ppp

Cn bng trn t2 vtac:21,

21,

21,1 DCBA

)1(2/1

)1()2/1(

)1(2/11

)1(2/1)2/1(

)1(2/11)(

22

2

ppp

pp

pp

pppY

Trang 8

Sdng bini ngc Laplace

Vy nghim ca phng trnh l: ttety t sin21

cos21

21

1)(

V d 3:

Tm nghim ca phng trnh vi phn:

0)0(''',0)0('',1)0(',0)0(0)4(

yyyyyy

Gii

Dng bini Laplace c2 v, tac:

1)(

)()1(

0)()(.

0)()0(''')0(''.)0('.)0(.)(.

4

2

24

24

234

pp

pY

ppYp

pYppYp

pYyypypyppYp

Dng phng phpi sphn tch vphi.

)1)(1)(1()()()()(

)1)(1)(1()1)(()1)(1()1)(1(

111)1)(1)(1(1

2

23

2

222

22

2

4

2

pppDBApCBApDBApCBA

ppppDCpppBppA

pDCp

pB

pA

pppp

pp

Cn bng t2 vtac:21

,0,41

,41 DCBA

12/1

14/1

14/1)( 2

ppppY

Sdng bini ngc Laplace:

Vy nghim phng trnh l: teety tt sin21

41

41)(

Trang 9

III. NG DNG Laplace GII PHNG TRNH VI PHN C V PHI LHM BC THANG:

Hm bc thang Heaviside:

1) nh ngha:

a)

0100

)(tt

tH

Hm bc thang Heaviside, cngc gi l hm bc thangn v, hm khng lin tc

ny nhn gi tr0 khii s(t) m v nhn gi tr1 khii s(t) dng. Hm nyc

sdng trong l thuyt ton hciu khin hay trong xl tn hiu.

b) V ta c hm tnh tin bc thang Heaviside

, cho sthc c. Ta c:

ctct

ctHtH c 10

)()(

Nu c>0 (c

Trang 10

V d: M thm:

ttt

tf12

102)(

sdng hm bc thang Heaviside.

Gii

T )(tf l hm khvi tng khc trn khong 10 t v 1t , chng ta sdng hm

khong H01(t) trn khong 10 t , v dng hm tnh tin H1(t) trn 1t .Vy:

)1()1(2)(.2

)1(2)1()(2)(2)(.2)( 101

tHttHt

tHtHtHttHtHttf

2) Bin i Laplace:

c

cpptpt

cc pe

dtedtetHpHL0

)()(

Chng minh:

p

eee

ppe

dtedtetHpHLcp

pcpb

b

b

c

pt

bc

ptptcc

1limlim)()(

0

( 1)( ctH khi ct v 0)( ctH khi t

Trang 11

0

)()()( defdtectf cpc

pt

)(

)(

)()()()(

0

0

)(

pFe

defe

defpctfctHL

cp

pcp

cp

V d:

Bin i Laplace cc hm sau:

a)

te

tt

tft 6

645403

)(7

Gii

Ta c:

)6(.)6(5)4(8)(3

)6()6(5)4(8)(3

)6()6()4(5)4()(3

)()(5)(3)(

)6(

7

7

67

4604

tHeetHtHtH

tHetHtHtH

tHetHtHtHtH

tHetHtHtf

t

t

t

t

Da vo cng thc bini Laplace ca hm bc thang Heaviside v bng bini

Laplace tac:

1

5.83)(

664

pe

ep

ep

ep

pfLppp

b)

2021sin

10)(

ttt

ttf

Trang 12

Gii

Ta c:

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

)2()1(sin

)(.0)(sin)(.0)( 2121

tHttHtttHttH

tHtHt

tHttHtHtf

Da vo cng thc bini Laplace ca hm bc thang Heaviside v bng bini

Laplace tac:

)()( 222

22

2

22

pee

pe

pe

pfLpppp

3) Bin i Laplace ngc:

Cho hm )(tf l hm lin tc tngon v )()( pfLpF th: )()()()(1 ctfctHtpFeL cp

Cc v d: NG DNG Laplace GII PHNG TRNH VI PHN C V PHI

L HM BC THANG

V d 1:

Tm nghim ca phng trnh vi phn:

5)0()('

y

tfyy

Khi:

ttt

tfcos3

00)(

Gii

Ta c:

1

3)(

)()cos(3)(cos3cos3)(.0)(

2

0

ppepfL

tHtttHtHtHtfp

Trang 13

Sdng tnh chto hm gc v bini Laplace tac:

)1)(1(3

)1(5)(

135)()1(

13)()0()(

2

2

2

pppe

ppY

ppe

pYp

ppepYyppY

p

p

p

M

p

pppp

epp

CApCBpBA

epp

pCBppAep

CBpep

App

pe

)1)(1()()(

)1)(1()1)(()1(

11)1)(1(

2

2

2

2

22

Cn bng 2 v: Cho21,

21

21 CBA

ppp

ppp

pppp

ep

ep

pe

pp

ep

ep

pe

pppY

ep

ep

pe

ppppe

11

111

23

)1(5

1)2/3(

1)2/3(

12/3

)1(5

)(

1)2/1(

1)2/1(

12/1

)1)(1(

22

22

222

Dng bini Laplace ngc tac:

)(cossin

235

)()cos()()sin()(235)(

)(

)(

tHttee

tHttHttHeety

tt

tt

Vy nghim phng trnh l:

tttee

tety

tt

t

cossin23

5

05)(

)(

Trang 14

V d 2:

Tm nghim ca phng trnh vi phn:

1)0(',0)0()(''

yytfyy

khi

ttt

tf12

102)(

Gii

22

1

11101

212)(

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

pe

ppfL

tHtttHtHtHtHttHttHtf

p

S dng tnh cht o hm gc viu kin u, bin i Laplace bn v tri l:

222 221)()1()())0(')0(.)(.('' pe

pYppYyyppYpyyLp

11

)1(22

)(

.221)()1(

222

22

pppe

pY

pepYp

p

p

M )1

2222(

)1(22

2222

pe

pe

ppe ppp

11

1222

11)

12222(

)1(22)(

2222

22222

ppe

pe

p

ppe

pe

ppepY

pp

ppp

Dng bin i Laplace ngc ta c:

)1()1sin()1(2sin2sin)1sin()1(2)1()1(22)(

tHtttt

tttHtHttty

Vy nghim phng trnh l:

ttt

tttty

1sin)1sin(2210sin2

)(

Trang 15

IV. NG DNG Laplace GII H PHNG TRNH VI PHN H S HNG

A. PHNG PHP CHUNG:

Cng nhphng trnh vi phn tuyn tnh hshng, gii hphng trnh vi phn

tuyn tnh hshng ta thay cc hm phi tm, cc o hm ca chng v cc hm v

phi (nu l hkhng thun nht) bng nh ca chng (bng cch p dngo hm gc).

Khi ta sthuc mt hphng trnhi stuyn tnh i vi nh ca cc hm phi

tm. Gii h v dng php bin i ngc tm gc, ta c nghim ring ca h

thomniu kin cho.

B. CC V D:

V d 1:

Tm nghim h phng trnh vi phn:

1)0(,1)0(

0'03'

yx

yxyyxx

Gii

Sdng tnh chto hm gc bini tac:

1)()1()(1)()()3(

0)()(1)(0)()(31)(

0)()()0()(0)()(3)0()(

pYppXpYpXp

pYpXppYpYpXppX

pYpXyppYpYpXxppX

Gii hphng trnhi s, tm nghim )(),( pYpX

2

2

)2(4)(

)2()(

pppY

pppX

Phn tch: 2,1)2(

)2()2()2()2( 222

BA

pBpA

pB

pA

pp

2D,1)2(

D)2()2()2()2(

4222

Cp

pCp

Dp

Cpp

Trang 16

2

2

)2(2

)2(1)(

)2(2

)2(1)(

pppY

pppX

Sdng bini Laplace ngc.

Vy nghim h phng trnh vi phn l:

teety

teetxtt

tt

..2)(

..2)(22

22

V d 2:

Tm nghim hphng trnh vi phn:

1)0(',0)0(,1)0(',0)0(

044''0410''

yyxx

yxyyxx

Gii

Sdng tnh chto hm gc bini tac

1)()4()(4

1)(4)()10(

0)(4)(4)0(')0()(

0)(4)(10)0(')0()(

2

2

2

2

pYppX

pYpXp

pYpXypypYp

pYpXxpxpXp

Gii hphng trnh theo phng phpi stac:

)12)(2(6)(

)12)(2()(

22

2

22

2

ppppY

ppp

pX

Trang 17

Phn tch:

53

',0',52

',0'

)12)(2('2'12)'2'12()''()''(

)12)(2()2)(''()12)(''(

)12(''

)2(''

)12)(2(6

56,0,

51,0

)12)(2(212)212()()(

)12)(2()2)(()12)((

)12()2()12)(2(

22

23

22

22

2222

2

22

23

22

22

2222

2

DCBA

ppDBCApDBpCAp

pppDpCpBpA

pDpC

pBpA

ppp

DCBA

ppDBCApDBpCAp

pppDCppBAp

pDCp

pBAp

ppp

)12

12(

1253

)2

2(

252

125/3

25/2

)(

)12

12(125

6)2

2(25

1125/6

25/1)(

2222

2222

pppppY

pppppX

Sdng bini Laplace ngc.

Vy nghim h phng trnh vi phn l:

ttty

tttx

32sin310

32sin

252

)(

32sin35

32sin25

1)(

V d 3:

Tm nghim h phng trnh vi phn:

1)0(,2)0(

2'

14''2

yx

tyxx

xyx

Trang 18

Gii

Sdng tnh chto hm gc tac:

)2(22

)()()2(

)1(11

)()()4(

2)(2)()2(

11)(2)()4(

2)()(2)0()(

1)(4)0()()0()(

3

3

3

ppYpXp

pppYpXp

ppYpXp

pppYpXp

ppYpXxppX

ppXyppYxppX

Nhn phng trnh (2) cho p, cng 2 phng trnh li tac:

)4()232(

)1)(4()232)(1(

)43(22

)(

22

11

)()2()()4(

2

2

2

2

22

23

22

pppp

pppppp

pppppp

pX

ppp

pXpppXp

Thay )(pX vo phng trnh (1), ta c:

)4(868

)4()232)(4()1)(4(

)(

)4()232)(4()1()(

)4()232)(4()1(

)(

11)()4(

)232()4(

3

23

3

22

3

2

2

2

2

2

2

ppppp

pppppppp

pY

ppppp

pppY

ppppp

pp

ppY

pppY

ppppp

Trang 19

)4(868

)(

)4()232()(

3

23

2

2

ppppp

pY

pppppX

Phn tch:

)4(4)4()(

)4()4()4(

4)4()232(

2

2

2

2

22

2

ppBpBApCA

ppCppBpAp

pC

pB

pA

pppp

Cn bng h s, ta c: 8

11,21,

85 CBA

)4(4)4()4()(

)4()4()4()4(

)4()4(868

3

23

3

32

323

23

ppCpCBpBApDA

ppDppCpBppAp

pD

pC

pB

pA

ppppp

Cn bng h s, ta c: 4

11,2,1,47 DCBA

)4(41121

47)(

)4(811

21

85)(

32

2

pppppY

ppppX

S dng bin i Laplace ngc

Vy nghim ca h phng trnh:

t

t

ettty

ettx

42

4

411

47)(

811

21

85

)(

Trang 20

V. KT LUN:

So vi phng php c in gii phng trnh vi phn hs hng ta thy phng

php sdng ton tLaplace c nhng uim sau:

-D n ln bao nhiu ta ch cn gii mt phng trnhi s bc nht i vi Y(p).

-Khi lng tnh ton ni chung t hn so vi phng php bin thin hng s Lagrange.

-Cho ngay nghim ring khng cn thng qua nghim tng qut. Trong trng hp mun

c nghim tng qut ch cn t y0 = C0, y0 = C1,, y0(n-1) = Cn-1. Vi Ck l nhng hng

s tu .