Phuong Trinh Vi Phan Cap 2

Trng i hc Bch khoa tp. H Ch MinhB mn Ton ng dng

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Gii tch 1

Chng 4.

Phng trnh vi phn tuyn tnh cp 2

H phng trnh vi phn tuyn tnh cp mt.

Ging vin Ts. ng Vn Vinh (11/2008)

[email protected]

Ni dung---------------------------------------------------------------------------------------------------------------------------

I Phng trnh vi phn tuyn tnh cp 2 tng qut.

III- H phng trnh vi phn tuyn tnh cp 1.

II Phng trnh vi phn tuyn tnh h s hng.

I. Phng trnh vi phn tuyn tnh cp 2

nh ngha phng trnh khng thun nht

Phng trnh vi phn tuyn tnh cp hai khng thun nht

'' '( ) ( ) ( ), (1)y p x y q x y f x

trong l cc hm lin tc.( ), ( ), ( )p x q x f x

nh ngha phng trnh thun nht

Phng trnh vi phn tuyn tnh cp hai thun nht

'' '( ) ( ) 0, (2)y p x y q x y

trong l cc hm lin tc.( ), ( )p x q x


0tq ry y y

Cu trc nghim ca phng trnh khng thun nht

l nghim tng qut ca pt khng thun nht.tqy

l nghim tng qut ca pt thun nht.0y

l nghim ring ca pt khng thun nht.ry

Tp hp cc nghim ca phng trnh thun nht l

khng gian 2 chiu: 0 1 1 2 2( ) ( )y c y x c y x

l nghim ring ca pt thun nht (2)1( )y x

' ' '

2 1 1 ;y y u y u

Tm nghim th hai dng: 2 1( ) ( )y y x u x

'' '' ' ' ''

2 1 1 12y y u y u y u

'' ' ' '' ' '1 1 111 1 02 y up qy u yy y uu yuu

'' ' '' ' '1 1 1 1 1 12 0y py qy u y u y py u '' ' '1 1 12 0y u y py u

t , c phng trnh tch bin'z u ' '1 1 12 0y z y py z ( )

21 ( )

p x dxeu dx

y x

( )

2 1 21

( ) ( )( )

p x dxey x y x dx

y x


Tm nghim ring ca (1) bng phng php bin thin

hng s:1 1 2 2( ) ( ) ( ) ( )ry c x y x c x y x

' ' ' ' '

1 1 1 1 2 2 2 2( ) ( ) ( ) ( ) ( ) ( )ry C x y C x y x C x y C x y x

'' '' ' ' ' ' ' '' '' ' ' ' ' ''

1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2ry C y C y C y C y C y C y C y C y

Thay vo pt (1):'' '( ) ( ) ( )r r ry p x y q x y f x

' '

1 1 2 2

' ' ' '

1 1 2 2

0

( )

C y C y

C y C y f x

Gii h tm .' '1 2,C C

Suy ra .1 2( ), ( )C x C x

Nghim ring: ry Nghim tng qut ca (1): 0tq ry y y

ch cn tm mt nghim ring ca pt thun nht.1( )y x

KT LUN:

gii phng trnh '' '( ) ( ) ( )y p x y q x y f x

T nghim suy ra:( )

2 1 2

1

( ) ( )( )

p x dxey x y x dx

y x

1( )y x

Tm nghim 1 1 2 2( ) ( ) ( ) ( )ry c x y x c x y x

' '

1 1 2 2

' ' ' '

1 1 2 2

0

( )

C y C y

C y C y f x

1 2( ), ( )C x C x ry

Nghim tng qut ca pt khng thun nht:0tq ry y y

V d Gii phng trnh 2 '' ' 34 (1)x y xy y x

Phng trnh thun nht: '' '2

1 10 (2)y y y

x x

on mt nghim ring ca pt thun nht: 1( )y x x

Tm nghim ring th hai ca (2):

( )

2 1 2

1

( ) ( )( )

p x dxey x y x dx

y x

1

2

dxxe

x dxx

Phng trnh chun: '' '2

1 14y y y x

x x

lnx x

Tm nghim ring ca pt (1) bng PP bin thin hng s

Trong bi ny ta on c: 3y x

Nghim tng qut ca (1): 0 1 23ln | |rtq y C xy C x x xy

V d Gii phng trnh '' 'tan 2 0y x y y

on mt nghim ring: 1( ) siny x x


( )

2 1 2

1

( ) ( )( )

p x dxey x y x dx

y x

tan

2sin

sin

xdxex dx

x

V d Gii phng trnh'' '

2 2

2 20

1 1

x yy y

x x

on mt nghim ring: 1( )y x x


( )

2 1 2

1

( ) ( )( )

p x dxey x y x dx

y x

2

2

1

2

xdx

xex dx

x

II. Ptrnh vi phn tuyn tnh cp 2 h s hng

nh ngha phng trnh khng thun nht h s hng

Phng trnh vi phn tuyn tnh cp hai l phng trnh

'' ' ( ), (1)y py qy f x

trong l hng s, v f(x) l hm lin tc.,p q

nh ngha phng trnh thun nht h s hng

Phng trnh vi phn tuyn tnh cp hai l phng trnh'' ' 0, (2)y py qy

trong l cc hng s.,p q

Gii phng trnh thun nht: '' ' 0, (2)y py qy

Phng trnh c trng: 2 0k pk q

TH 1: PTT c hai nghim thc phn bit1 2,k k

TH 2: PTT c mt nghim kp0k

Nghim tng qut: 1 20 1 2

k x k xy C e C e

Nghim tng qut: 0 00 1 2

k x k xxy C e C e

TH 3: PTT c mt nghim phc1k a bi

Nghim tng qut: 0 1 2cos sinaxy e C bx C bx

Tm nghim ring ca phng trnh khng thun nht

Trng hp chung: Phng php bin thin hng s.

Xt hai trng hp c bit:

TH 1: , Pn(x) l a thc bc n.( ) ( )x

nf x P x e

Tm dng: ( )s x

r ny x e Q xry

s = 0, nu khng l nghim ca pt c trng.

s = 1, nu l nghim n ca pt c trng.

s = 2, nu l nghim kp ca pt c trng.

l a thc cng bc vi cc h s cn tm.( )nP x( )nQ x

tm cc h s ny, thay vo pt (1).ry

TH 2: ( ) ( )cos ( )sinx n mf x e P x x Q x x

Tm dng: ( )cos ( )sins xr k ky x e H x x T x x ry

s = 0, nu khng l nghim ca pt c trng.i

s = 1, nu l nghim n ca pt c trng.i

: hai a thc bc vi cc h s cn tm.max{ , }k m n,k kH T

tm cc h s ny, thay vo pt (1):ry

'' ' ( )r r ry py qy f x

V sinx v cosx c lp tuyn tnh nn cc h s tng

ng bng nhau.

Ch :1) C nguyn l cng dn (chng cht) nghim:

'' '

1 2( ) ( ) ( )y py qy f x f x f x

nghim ring ca (1) c dng1 2r r r

y y y

nghim ring ca pt:1r

y '' '1( )y py qy f x

nghim ring ca pt:2r

y '' '2( )y py qy f x

2) l trng hp 1:( ) ( )nf x P x0( ) ( )x nf x e P x

3) l trng hp 2:( ) ( )cosnf x P x x

0( ) ( )cos 0sinx nf x e P x x x

V d Gii phng trnh '' '5 6 xy y y e

( )s xr ny x e Q x

Phng trnh c trng: 2 5 6 0k k 1 22 3k k

Nghim tng qut ca pt thun nht: 2 30 1 2x xy C e C e

( ) ( )x xnf x e P x e bc 0.1, ( )nP x

1, ( )nQ x A (v Pn bc 0)

s = 0, v khng l nghim pt c trng.1

0 x x

ry x e A Ae ' '',x xr ry Ae y Ae

'' '5 6 xr r ry y y e 5 6x x x xAe Ae Ae e

1

12A

Klun: Nghim t/qut: 0tq ry y y 2 3

1 2

1

12

x x xC e C e e

V d Gii phng trnh '' 24y y x

( )s xr ny x e Q x

Phng trnh c trng: 2 4 0k 1 22 2k i k i

Nghim t/qut ca pt th/nht: 00 1 2cos2 sin 2xy e C x C x

2 0( ) ( ) xnf x x P x e bc 2.0, ( )nP x

20, ( )nQ x Ax Bx C (v Pn bc 2)

s = 0, v khng l nghim pt c trng.0

0 0 2 2xry x e Ax Bx C Ax Bx C ' ''2 , 2r ry Ax B y A

'' 24r ry y x 2 22 4( )A Ax Bx C x

1 1, 0,

4 8A B C

Nghim t/qut: 0tq ry y y 2

1 2

1 1cos2 sin 2

4 8C x C x x

V d Gii phng trnh '' '2 3y y x

( )s xr ny x e Q x

Phng trnh c trng: 2 2 0k k 1 20 2k k

Nghim t/qut ca pt th/nht:0 2

0 1 2

x xy C e C e

0( ) 3 ( ) xnf x x P x e bc 1.0, ( )nP x

0, ( )nQ x Ax B (v Pn bc 1)

s = 1, v l nghim n pt c trng.0

1 0 2xry x e Ax B Ax Bx ' ''2 , 2r ry Ax B y A

'' '2 3ry y x 2 2(2 ) 3A Ax B x 3 3

,4 4

A B


1 2

3 3

4 4

xC C e x

V d Gii phng trnh '' '2 2 xy y y e

( )s xr ny x e Q x


Nghim t/qut ca pt th/nht: 0 1 2x xy C e C ex

1( ) 2 ( )x xnf x e P x e bc 0.1, ( )nP x

1, ( )nQ x A (v Pn bc 0)

s = 2, v l nghim kp pt c trng.1

22 x x

ry x e A Ax e ' 2 '' 2(2 ), (2 4 )x xr ry Ae x x y Ae x x

'' '2 2 xr r ry y y e 3 3

,4 4

A B


1 2

3 3

4 4

xC C e x

V d Gii phng trnh '' '4 3 sin 2y y y x

( )cos2 ( )sin 2s xr k ky x e T x x H x x

Nghim t/qut ca pt th/nht:3

0 1 2

x xy C e C e

0( ) (0.cos2 sin 2 )xf x e x x

bc 0.0, 2, ( ), ( )n mP x Q x

0, ,k kT A H B

s = 0, v khng l nghim pt c trng.2i i

Phng trnh c trng: 2 4 3 0k k

max{ , } 0k m n

1 21 3k k

cos2 sin 2ry A x B x

' ''2 sin 2 2 cos2 , 4 cos2 4 sin 2r ry A x B x y A x B x

'' '4 3 sin 2r r ry y y x


1 2

8 1cos2 sin 2

65 65

x xC e C e x x

cos2 si

2 si4

3 si

n 2

n 2

4 cos

n

2 2 co

2

4 in 2 s2s A x B

A x

A xx B x

B x x

( 8 )cos2 (8 )sin 2 sin 2A B x A B x x

8 0

8 1

A B

A B

8 1,

65 65A B

8 1cos2 sin 2

65 65ry x x

( )cos ( )sins xr k ky x e T x x H x x

Nghim t/qut ca pt th/nht: 00 1 2cos sinxy e C x C x

0( ) (cos 0.sin )xf x e x x

bc 0.0, 1, ( ), ( )n mP x Q x

0, ,k kT A H B

s = 1, v l nghim pt c trng.i i

Phng trnh c trng: 2 1 0k

max{ , } 0k m n

1 2k i k i

V d Gii phng trnh '' cosy y x

cos sinry x A x B x ' ( )cos ( )sinry A Bx x B Ax x

'' cosr ry y x

Nghim t/qut: 0tq ry y y 1 21

cos sin sin2

C x C x x x

-2 - sin 2 - cos cos s oin c sA Bx x B Ax x x xA x B x

10,

2A B

1sin

2ry x x

'' -2 - sin 2 - cosry A Bx x B Ax x

2 sin 2 cos cosA x B x x

( )cos ( )sins xr k ky x e T x x H x x

Nghim t/qut pt th/nht: 0 1

1

22 cos

7 7

2n

2si

x

y e C x C x

0( ) (cos 2sin )xf x e x x

bc 0.0, 1, ( ), ( )n mP x Q x

0, ,k kT A H B

s = 0, v khng l nghim pt c trng.i i

Phng trnh c trng: 2 2 0k k

max{ , } 0k m n

1

1

2

7

2k i

V d Gii phng trnh '' ' 2 2sin cosy y y x x

cos sinry A x B x ' sin cosry A x B x

'' ' 2 cos 2sinr r ry y y x x

Nghim t/qut: 0tq ry y y

1

21 2

7 7 3 1cos sin cos sin

2 2 2 2

x

tqy e C x C x x x

sin cos cos sincos si 2 c s 2 nn o siA x B xA A x xx B x xB x

1

2

A B

A B

1 3,

2 2A B

'' cos sinry A x B x

( )cos ( )sin cos 2sinA B x A B x x x

1 3cos sin

2 2ry x x

Nghim t/qut pt th/nht: 2 20 1 2x xy C e C xe

2

1 2( ) ( ) ( )xf x f x f x x e


V d Gii phng trnh'' ' 24 4 xy y y x e

S dng nguyn l cng dn nghim

Tm nghim ring ng vi :1( )f x

'' '

14 4 ( ) (1)y y y x f x

1( )s xr ny x e Q x

0, ( )nQ x Ax B (v Pn bc 1)

s = 0, v khng l nghim n pt c trng.0

Thay vo pt (1), ta c1r

y Ax B 1

4A B

1

1 1

4 4ry x

Thay vo pt (2), ta c1

2A

Tm nghim ring ng vi :2( )f x

'' ' 2

24 4 ( ) (2)xy y y e f x

2

2 2x

ry x e A

1( )s xr ny x e Q x

2, ( )nQ x A (v Pn bc 0)

s = 2, v l nghim kp pt c trng2

Mt nghim ring ca bi l:

1 2r r ry y y 2 2

1 1 1

4 4 2

xx x e

Nghim t/qut: 0tq ry y y

2 2 2 21 2

1 1 1

4 4 2

x x xtqy C e C xe x x e

II. H pt vi phn tuyn tnh cp 1 h s hng.

H phng trnh vi phn (n phng trnh, n hm s)

111 1 12 2 1 1

221 1 22 2 2 2

1 1 2 2

... ( )

... ( )

... ( )

n n

n n

nn n nn n n

dxa x a x a x f t

dt

dxa x a x a x f t

dt

dxa x a x a x f t

dt

(1)

trong l cc hm theo t, lin tc.( )f t

l cc hm theo t.1 2( ), ( ), , ( )nx t x t x t

II. H pt vi phn tuyn tnh cp 1 h s hng

11 12 1

21 22 2

1 2

n

n

n n nn

a a a

a a aA

a a a

1

2

n

x

xX

x

1

2

( )

( )( )

( )n

f t

f tF t

f t

H phng trnh dng ma trn: ( )dX

AX F tdt

(2)

H phng trnh thun nht:dX

AXdt

(3)

Nghim ca h l hm vct trn khong (a,b) c to

l cc hm kh vi lin tc trn (a,b) v tho h:

II. H phng trnh vi phn tuyn tnh cp 1

Cu trc nghim ca h tuyn tnh (2)

0tq rX X X

l nghim tng qut ca h pt khng thun nht (2)tq

X

l nghim tng qut ca h pt thun nht (3)0

X

l nghim ring ca h pt khng thun nht (2)r

X

Phng php kh

Ni dung phng php kh l a h phng trnh vi

phn v phng trnh vi phn cp cao hn bng cch

o hm mt phng trnh ri kh cc hm cha bit.

u im

Gii h phng trnh rt nhanh.

Nhc im

Rt kh gii h nhiu phng trnh, nhiu hm.

V d Gii h phng trnh

'

1 1 2

'

2 1 2

2

4 3

x x x

x x x

Ly phng trnh (2) tr 4 ln phng trnh (1).

' '

1 2 24 5 (*)x x x

o hm hai v phng trnh (2).

'' ' '

2 1 24 3 x x x ' ' ''

1 2 24 3 x x x Thay vo pt (*)

Thay vo pt 2 ca h

' '' '

2 2 2 23 5x x x x '' '

2 2 24 5 0x x x

5

1 1 2( )t tx t C e C e

5

2 1 2

1( )

2

t tx t C e C e

V d Gii h phng trnh

'

1 1 2

'

2 1 2

3

2 2

tx x x e

x x x t


' '

1 2 12 4 2 (*)tx x x t e

o hm hai v phng trnh (1).

'' ' '

1 1 23 tx x x e ' '' '2 1 13

tx x x e Thay vo pt (*)

' '' '

1 1 1 12 3 4 2t tx x x e x t e

'' '

1 1 15 4tx x x t e

4

1 1 2

5( )

9 3 4 16

t tt t e te tx t C e C e Thay vo pt 1 ca h

4

2 1 2

8 2 3 11( ) 2

9 3 4 16

t tt t e te tx t C e C e

V d Gii h phng trnh

'

1 1 2 3

'

2 1 2 3

'

3 1 2 3

3

2 4 2

3

x x x x

x x x x

x x x x


' '

1 2 1 34 10 2 (*)x x x x

Ly phng trnh (3) tr phng trnh (1).

' '

1 3 1 32 2 (**)x x x x

o hm hai v pt (1): ' '' ' '2 1 1 33x x x x

Thay vo pt (*):' '' ' '

1 1 1 3 1 34 3 10 2 x x x x x x

'' ' '

1 1 3 1 37 10 2 (***)x x x x x

Cng hai pt (**) v (***)'' '

1 1 18 12 0x x x

6 2

1 1 2( )t tx t C e C e

6 2

3 1 3( )t tx t C e C e

Thay vo pt (**):

' 6

3 3 12 4tx x C e

Thay vo pt (1) ca h, ta c:'

2 1 1 33x x x x

6 2 6 2 6 2

2 1 2 1 2 1 3( ) 6 2 3 3t t t t t tx t C e C e C e C e C e C e

6 22 1 2 3( ) 2t tx t C e C C e

Nghim ca h cho:

1

2

3

( )

( )

( )

x t

x t

x t

V d Gii h phng trnh

'

1 1 2 3

'

2 1 2 3

'

3 1 2 3

6 12

3

4 12 3

x x x x

x x x x

x x x x

Ly phng trnh (2) tr phng trnh (1).

' '

1 2 1 25 9 (1)x x x x

Ly pt th ba ca h cng 3 ln pt u ca h

' '

1 3 1 23 14 24 (2)x x x x

o hm hai v pt (2): ' ' ' ''3 1 2 23x x x x

Thay vo pt (2):' ' ''

1 2 2 1 24 3 14 24 (3)x x x x x

Kh trong pt (1) v (3):1x' ' ''

1 2 2 26 5 6 (4)x x x x

o hm hai v (5):

Kh trong pt (1) v (3):'1x

' ''

2 2 1 26 12 (5)x x x x

'' ''' ' '

2 2 1 26 12 (6)x x x x

Rt thay vo (4):'1x

''' '' '

2 2 2 26 12 6 0x x x x

Gii phng trnh ny ta c 2 32 1 2 3( )t t tx t C e C e C e

Thay vo (4) ta c 1( )x t

Thay vo u ca h ta c 3( )x t

V d Gii h phng trnh

'

1 1 2 3

'

2 1 2 3

'

3 1 2 3

2 4 3

4 6 3

3 3

x x x x

x x x x

x x x x

Cng hai phng trnh u ca h.

' '

1 2 1 22 2 (1)x x x x

Ly pt u tr 3 ln pt u ca h

' '

1 3 1 23 7 5 (2)x x x x

o hm hai v pt u: ' ' ' ''3 1 2 13 2 4x x x x

Thay vo pt (2):' ' ''

1 2 1 1 23 4 7 5 (3)x x x x x

' ' ''

1 2 1 13 2 4 (4)x x x x

o hm hai v (5):

Kh trong pt (1) v (3):2x

Kh trong pt (1) v (3):'2x

' ''

1 1 1 23 (5)x x x x

'' ''' ' '

1 1 1 23 (6)x x x x

Rt thay vo (4):'2x

''' ''

1 1 13 4 0x x x

Gii phng trnh ny ta c 2 21 1 2 3( )t t tx t C e C e C te

Thay vo (4) ta c 2 ( )x t

Thay vo u ca h ta c 3( )x t

V d Gii h phng trnh

'

1 1 2

'

2 2 3

'

3 1 2 3

3

2 2

tx x x e

x x x t

x x x x t

Ly 3 ln pt u tr pt th hai ca h.

' '

1 2 1 33 3 3 - (1)tx x x x e t

Ly pt u tr pt th 3 ca h

' '

1 3 1 32 2 - 2 (2)tx x x x e t

o hm hai v pt u: ' ' ''2 1 1

tx x x e

Thay vo pt (1):' ''

1 1 1 34 3 2 - (3)tx x x x e t

' '' '

1 1 3 19 2 8 5 - 4 (4)tx x x x e t

o hm hai v (3):

Kh trong pt (2) v (3):3x

'' ''' ' '

1 1 1 34 3 2 -1 (5)tx x x x e

Rt thay vo (4):'3x

''' '' '

1 1 1 16 12 8 3 4 1tx x x x e t

Gii phng trnh ny ta c

2 2 2 2

1 1 2 3

5( ) 3

2 8

t t t t tx t C e C te C t e e

Thay vo pt u ca h ta c 2 ( )x t

Thay vo pt hai ca h ta c 3( )x t

Phng php tr ring, vct ring

A l ma trn thc, vung cp n.

Trng hp 1: A cho ho c:

( ) (2)dX

AX F tdt

1A PDP ( )dX

AX F tdt

1 ( )dX

PDP X F tdt

1 1 1 ( )dX

P DP X P F tdt

1Y P Xt ' 1 'Y P X

' 1 ( )Y DY P F t Ta c:

y l cc phng trnh vi phn cp 1 tch ri nhau.

V d Gii h phng trnh

'

1 1 2

'

2 1 2

3

2 2

tx x x e

x x x t

Cho ho A:

1

2

xX

x

3 1

2 2A

( )te

F tt

1 1 1 4 0 2/3 1/3

1 2 0 1 1/3 1/3A PDP

1Y P Xt ' 1 ( )Y DY P F t Ta c:

'1 1

'22

4 0 2 /3 1/3

0 1 1/3 1/3

ty y e

y ty

'1 1

'22

4 0 2 /3 1/3

0 1 1/3 1/3

ty y e

y ty

'1 1

'2 2

24

3 3

1

3 3

t

t

ty y e

ty y e

h gm hai ptrnh vi phn

tuyn tnh cp 1 ring bit

41 1

2 2

2 1( )

9 12 48

1( )

3 3 3

t t

t t

ty t C e e

t ty t C e e

Nghim ca h: 1 1

2 2

x yP

x y

1

2

1 1

1 2

y

y

V d Gii h phng trnh

'

1 1 2 3

'

2 1 2 3

'

3 1 2 3

3 4

2 4 2

3 8

x x x x t

x x x x

x x x x

Cho ho A ( Xem i s tuyn tnh)

1

2

3

x

X x

x

3 1 1

2 4 2

1 1 3

A

4

( ) 0

8

t

F t

1

1 1 1 2 0 0 1/ 2 1/ 2 1/ 2

1 0 2 0 2 0 1/ 4 1/ 4 3/ 4

0 1 1 0 0 6 1/ 4 1/ 4 1/ 4

A PDP


'

1 1

'

2 2

'

3 3

2 0 0 1/ 2 1/ 2 1/ 2 4

0 2 0 1/ 4 1/ 4 3/ 4 0

0 0 6 1/ 4 1/ 4 1/ 4 8

y y t

y y

y y

'1 1

'2 2

'3 3

2 2 4

2 6

6 2

y y t

y y t

y y t

21 1

22 2

63 3

( ) 5/ 2

( ) / 2 11/ 4

( ) / 6 19 /36

t

t

t

y t C e t

y t C e t

y t C e t

Nghim

ca hX PY

211

22 2

63 3

5/ 2( ) 1 1 1

( ) 1 0 2 / 2 11/ 4

( ) 0 1 1 / 6 19 /36

t

t

t

C e tx t

x t C e t

x t C e t

Phng php tr ring, vct ring

Trng hp 2: A khng cho ho c:

( ) (2)dX

AX F tdt

1(2) ( )dX

PTP X F tdt

1 1 1 ( )dX

P TP X P F tdt

1Y P Xt ' 1 'Y P X

' 1 ( )Y TY P F t Ta c:

Mi ma trn (thc hoc phc) u tam gic ho c.

1A PTP vi T l ma trn tam gic.

y l h tam gic,

gii t di ln.

V d Gii h phng trnh

'

1 1 2 3

'

2 1 2 3

'

3 1 2 3

2 4 3

4 6 3

3 3

x x x x

x x x x

x x x x

A khng cho ho c ( Xem i s tuyn tnh)

1

2

3

x

X x

x

2 4 3

4 6 3

3 3 1

A

y l h thun nht.

2| | 0 ( 2) ( 1) 0A I

1 2 c mt VTR c lp tuyn tnh 1

1

1

0

X


1

1

1

X

Tm hai ma trn1

2

3

1 1

1 1

0 1

x

P x

x

2 0

0 2 0

0 0 1

m

T

1A PTP AP PT 2 1 22AX mX X

Gi l ct th hai ca ma trn P.2X

Chn m = 1

1 1

2 2

3 3

2 4 3 1

4 6 3 1 2

3 3 1 0

x x

x x

x x

1

2

3

1

1

x

x

x

Chn 1 2

2

1

1

X

1 2 1

1 1 1

0 1 1

P

2 1 0

0 2 0

0 0 1

T

1Y P Xt 'Y TYTa c:

'

1 1

'

2 2

'

3 3

2 1 0

0 2 0

0 0 1

y y

y y

y y

'1 1 2

'2 2

'3 3

2

2

y y y

y y

y y

2 21 1 2

22 2

3 3

( )

( )

( )

t t

t

t

y t C e C te

y t C e

y t C e

1 1

2 2

3 3

( ) 1 2 1

( ) 1 1 1

( ) 0 1 1

x t y

x t y

x t y

V d Gii h phng trnh

'

1 1 2

'

2 2 3

'

3 1 2 3

3

2 2

tx x x e

x x x t

x x x x t

A khng cho ho c ( Xem i s tuyn tnh)

1

2

3

x

X x

x

1 1 0

0 3 1

1 1 2

A

3| | 0 ( 2) 0A I


1

1

1

X

( )

2

te

F t t

t

Tm hai ma trn1 1

2 2

3 3

1

1

1

x y

P x y

x y

2

0 2

0 0 2

a b

T c

1A PTP AP PT 2 1 22AX aX X

Gi l ct th hai ca ma trn P.2X

Chn a = 1

1 1

2 2

3 3

1 1 0 1

0 3 1 1 2

1 1 2 1

x x

x x

x x

1

2

3

1

1

x

x

x

Chn 2 2

1

2

1

X

1

2

3

1 1

1 2

1 1

y

P y

y

2 1

0 2

0 0 2

b

T c

1A PTP AP PT 3 1 2 32AX bX cX X Chn b = 0,c=1

1 1

2 2

3 3

1 1 0 1

0 3 1 2 2

1 1 2 1

y y

y y

y y

1

2

3

1

2

x

x

x

Chn 2 3

1

2

0

X

1 1 1

1 2 2

1 1 0

P

2 1 0

0 2 1

0 0 2

T

1

2 1 0

2 1 1

1 0 1

P

'1 1

'

2 2

'

3 3

2 1 0 2 1 0

0 2 1 2 1 1

0 0 2 1 0 1 2

ty y e

y y t

y y t

'1 1 2

'2 2 3

'3 3

2 2

2 2 3

2 2

t

t

t

y y y e t

y y y e t

y y e t

1Y P Xt ' 1 ( )Y TY P F t Ta c:

1

22 2 3

3 3

( )

( ) 2 / 2 3/ 4

( ) 2 2 2

t t t

t t

y t

y t C e C e te t

y t C e te t

Nhn xt:

sau khi kh ta c phng trnh vi phn tuyn tnh

cp cao ca mt pt. Phng trnh c trng ca pt ny

trng vi pt c trng ca ma trn A, hoc trong mt s

Gii h bng phng php kh:' ( )X AX F t

trng hp trng vi phng trnh ti thiu caA.

Phng php kh: 1) Kh ln lt tng bin trong h.

2) trong qu trnh kh: o hm hai v.

H 3 pt, 3 n: kh d dng, h nhiu pt nhiu n: kh

Ni dung n tp------------------------------------------------------------------------------------------------

I) Gii hn v lin tc: Cch tm gii hn hm, lin tc hm s.

II) o hm v vi phn: o hm v vi phn ca hm y = f(x),

hm tham s, hm n. Cng thc Taylor, Maclaurint.

ng dng o hm: cc bi ton lin quan, kho st v.

III) Tch phn: 1) Tch phn bt nh, tch phn xc nh

Tch phn suy rng loi mt v hai: tnh tphn, kho st hi t.

IV) phng trnh vi phn:

1) Phng trnh vi phn cp 1: ch c 5 loi hc: Tch bin, tuyn

tnh, ng cp, ton phn (khng c tha s tch phn), Bernoulli.

3) H phng trnh vi phn tuyn tnh cp 1 h s hng: Phng

php kh, phng php tr ring, vct ring (trng hp cho c)

2) Phng trnh vi phn cp hai H S HNG.

ng dng hnh hc ca tch phn: c 4 ng dng hc.

mu cui k 1------------------------------------------------------------------------------------------------

Cu 1. Tnh

arcsin

30

coshlim

tan 1 3 2cos 1

x

x

x x e

x x x

Cu 3. Tnh tch phn

3 22 2 2

dxI

x x x

Cu 2. Tm tim cn ca ng cong cho bi ptrnh tham s

2 2

8 3,

4 ( 4)

tx y

t t t

21

3 30

cosh ln(1 ) 1

2 8

x xI dx

x

Cu 4. Tnh tt c tch phn sau hi t.

Cu 6. Gii phng trnh vi phn 'y y x y

Cu 5. Tm th tch vt th trn xoay khi quay min D gii hn bi

arctan , arctan ,0 1, 1y x x y x x x x quanh trc Ox.

Cu 7. Gii phng trnh vi phn cp 2

'' '3 2 3 5sin 2y y y x x

Cu 8. Gii h phng trnh vi phn bng phng php tr ring.'

1 1 2 3

'

2 1 2 3

'

3 1 2 3

4 4

8 11 8 2

8 8 5

tx x x x e

x x x x t

x x x x

Cui k thi T LUN (trnh by cn thn), thi gian: 90pht.

Gii mu 1.

Cu 1.

3 3

3 30

/ 3 ( ) 1lim

44 / 3 ( )x

x xI

x x

22cosh 1 ( )

2

xx x

33arcsin ( )

6

xx x x

2 3arcsin 31 ( )

2 3

x x xe x x 3

3tan ( )3

xx x x

32 33 51 3 1 ( )

3

xx x x x

Cu 2. Tim cn ng: 0

0

lim ( )

lim ( ) 2

t

t

y t

x t

2x l tim cn ng.

Khng c tim cn ngang.

Tim cn xin.2

2

lim ( )

lim ( )

t

t

y t

x t

1

4 8

xy

l tim cn xin.

2

( ) 1lim

( ) 4t

y ta

x t

2

1lim ( ) ( )

8tb y t a x t

2

2

lim ( )

lim ( )

t

t

y t

x t

3 9

20 40

xy l tim cn xin.

2

( ) 3lim

( ) 20t

y ta

x t

2

9lim ( ) ( )

40tb y t a x t

Cu 3.1

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

A B C

x x x x x x

tm A, nhn 2 v cho x - 1 ri thay x = 1 vo, tng t cho B, C.

3 22 2 2

dxI

x x x

22 2cosh ln(1 ) 1 ( )

2

xx x x

Cu 4.

2

1 1 1ln | 1| ln | 1| ln | 2 |

6 2 3x x x

1/ 6 1/3

1/ 2

2

1 2ln

( 1)

x x

x

ln3 2ln 2

2 3

30

3 32 812

x xx

2 2

33 3

cosh ln(1 ) 1 12( )

22 8

x x xf x

xx

Hm di du tch phn l hm lun m. Xt tch phn hm - f(x).

3 2

12 1

2 x

Tch phn hi t nu 3 2 1 1

Cu 5. 1 2arctan , arctany x x y x x

Ta c 1 20 ( ) ( ), [0,1]y x y x x

1

2 2

0 2 1

0

xV y y dx 1

2 2

0

( arctan ) ( arctan )x x x x dx

1

0

0

4 arctanxV x xdx 2 2

Cu 6. y l phng trnh Bernoulli vi

'y y x y

1/ 2

'

2 22

y y x

y t z y

'

2 2

z xz / 2 2xz Ce x

Nghim tng qut ca pt l/ 2 2xy Ce x

Cu 7. Phng trnh c trng: 2 3 2 0k k 1 21 2k k

Nghim ca phng trnh thun nht:2

0 1 2

x xy C e C e

Dng nguyn l cng dn nghim tm nghim ring:1 2r r r

y y y

Nghim ring ca l'' '3 2 3y y y x

1

3 9

2 4ry x

Nghim ring ca :'' '3 2 5sin 2y y y x 2

3 1cos2 sin 2

4 4ry x x

Nghim tng qut ca phng trnh cho:

2

1 2

3 9 3 1cos2 sin 2

2 4 4 4

x xtqy C e C e x x x

Cu 8.

1 4 4

8 11 8

8 8 5

A

1 1 1

2 0 1

2 1 0

P

1 0 0

0 3 0

0 0 3

D


'

1 1

'

2 2

'

3 3

1 0 0 1 4 /3 4 /3

0 3 0 8/3 3 8/3 2

0 0 3 8/3 8/3 7 /3 0

ty y e

y y t

y y

'

1 1

'

2 2

'

3 3

8 / 3

3 8 / 3 6

3 8 / 3 16 / 3

t

t

t

y y e t

y y e t

y y e t

1 1

3

2 2

3

3 3

( ) 8 / 3 8/ 3

( ) 2 / 3 2 2 / 3

( ) 2 / 3 16 / 9 16 / 27

t t

t t

t t

y t C e te t

y t C e e t

y t C e e t

Suy ra nghim tng qut ca h.

mu cui k 2------------------------------------------------------------------------------------------------

Cu 1. Tnh

2

0

sinh ln(1 )lim

tanx

x x

x x

Cu 3. Tnh tch phn

2

21 3 2 1

dxI

x x x

Cu 2. Tm tim cn ca ng cong cho bi ptrnh44 1

| |

xy

x

3 22

ln

1arctan

xI dx

xx

Cu 4. Tnh tt c tch phn sau hi t.,

Cu 6. Gii phng trnh vi phn ' 1 1y x y x y

Cu 5. Tm din tch b mt trn xoay khi quay min D gii hn bi

( sin ), (1 cos ),0 2 ; 0x a t t y a t t a quanh trc Oy.

Cu 7. Gii phng trnh vi phn cp 2

'' ' 24 4 cosxy y y e x

Cui k thi T LUN (trnh by cn thn), thi gian: 90pht.

Cu 8. Gii h phng trnh vi phn bng phng php kh.' 2

1 1 2 3

'

2 1 2 3

'

3 1 2 3

4 2 5

6 6 2

8 3 9

x x x x t

x x x x t

x x x x

Gii mu cui k 2------------------------------------------------------------------------------------------------

Cu 1.2 3 3sinh ln(1 ) ( )x x x x

33tan ( )

3

xx x x

2 3

30 0

sinh ln(1 )lim lim 3

tan /3x x

x x x

x x x

Cu 2.

( )limx

f xa

x

Tim cn ng: 0x

44 1lim

| |x

x

x x

2,

2,

x

x

2a lim ( )x

b f x ax

44 1

lim 2x

xx

x

0

2a lim ( )x

b f x ax

0

C hai tim cn xin: 2 , 2y x y x

Cu 3. t2

1 2

2

2 13

dxI

xx x

1t

x

1

21/ 2 3 2

dt

t t

1

21/ 2

( 1)

4 ( 1)

d tI

t

1

21/ 2

( 1) / 2

1 ( 1) / 2

d t

t

1

1/ 2

1arcsin

2

t

3arcsin

2 4

Cu 4.2 /3

3 2

ln ln( )

1arctan

xx x xf x

xx

x

2 /3

ln x

x

Nu , th tch phn hi t vi mi2 /3 1 1/3

Nu , th tch phn phn k vi mi2 /3 1 1/3

Nu , th tch phn hi t khi2 /3 1 1/3 1

Cu 5. ( sin ), (1 cos ),0 2x a t t y a t t

'( ) cosx t a a t '( ) siny t a t

2 2

' ' 2 2( ) ( ) 4 sin2

tx t y t a

2

2 2' '

0

0

2 ( ) ( ) ( )yS x t x t y t dt

22 2

0

0

2 ( sin ) 4 sin2

y

tS a t t a dt

2

2

0

4 ( sin ) sin2

ta t t dt

22

0

0

4 sin sin sin2 2

y

t tS a t t dt

2 216 a

Cu 6.' 1

1

x yy

x y

t 1u x y

' '1u y

' 21u

uu

2du u u

dx u

2

udu dx

u u

2 82 ln 1 ln 2

3 3u u u x C

2 81 ln 1 1 ln 1 2

3 3x y x y x y x C

Cu 7. Nghim tng qut ca pt thun nht:2 2

0 1 2

t ty C e C te

Dng nguyn l cng dn nghim tm nghim ring:1 2r r r

y y y

Nghim ring ca :'' ' 24 4 xy y y e

1

2 21

2

xry x e

Nghim ring ca :'' '4 4 cosy y y x 2

3 4cos sin

25 25ry x x

Nghim tng qut ca phng trnh cho:

2 2 2 2

1 2

1 3 4cos sin

2 25 25

x x xtqy C e C xe x e x x

Cu 8.' 2

1 1 2 3

'

2 1 2 3

'

3 1 2 3

4 2 5

6 6 2

8 3 9

x x x x t

x x x x t

x x x x

Ly pt u cng vi 2 ln pt th hai ca h

' ' 2

1 2 1 32 8 7 4 (1)x x x x t t

Ly 3 ln pt u tr 2 ln pt th ba ca h

' ' 2

1 3 1 33 2 4 3 2 (2)x x x x t

Kh pt (2) v (3):1x'' ' ' 2

3 1 3 32 6 3 -9 12 (4)x x x x t t

o hm hai v (5):

Kh pt (2) v (3):'1x

'' ' 2

3 3 1 36 16 4 6 - 27 36 (5)x x x x t t

''' ''' ' ' 2

3 3 1 36 16 4 - 6 -54 36 (6)x x x x t

Rt thay vo (4):'1x

''' '' ' 2

3 3 3 34 5 2 3 8 6x x x x t t

Gii pt ny:2

2 3

3 1 2 3

3 23 79( )

2 2 4

t t t t tx t C e C e C e

Thay vo (4) ta c 1( )x t

Thay vo u ca h ta c 2 ( )x t

o hm hai v pt th 3: ' '' ' '2 3 1 31

8 9 3

x x x x

'' ' ' 2

3 1 3 1 32 19 18 24 21 3 12 (3)x x x x x t t Thay vo (1):

Bi tp

'' '1) 2 2 xy y y e

'' '2) 7 6 sin y y y x

'' '3) 2 2 2 y y y x

'' ' 24) 3 2 3 xy y y e

'' '5) 7 6 (3 4 ) xy y y e x

'' ' 26) 5 5 2 1 2y y x x

'' '7) 5 29 sin 2y y x x

/ 2

1 2

x x xy C e C e e

6

1 2

7 5cos sin

74 74

x xy C e C e x x

1 2cos sin 1xy e C x C x x

2 2

1 2 (3 3)x x xy C e C e x e

'' '8) 2 5 100 cos xy y xe x

'' ' 29) 4 4 3 xy y y e

'' '10) 4 4 sin cos2 y y y x x

'' ' 311) 4 4 sin y y y x

'' '12) 4 4 sinh 2 y y y x

''13) cos y y x

''14) sin 2 xy y x e

'' '15) 6 5 cosh 5 xy y y e x

Phuong Trinh Vi Phan Cap 2

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Transcript of Phuong Trinh Vi Phan Cap 2