01 Slide - Chuong 4 Phuong Trinh Vi Phan

7/24/2019 01 Slide - Chuong 4 Phuong Trinh Vi Phan

1/31

VI TCH PHN A2

Chng 5. PHNG TRNH VI PHN

CBGD. L Hoi Nhn

Ngy 10 thng 11 nm 2014

CBGD. L Hoi Nhn () Chng 5. PHNG TRNH VI PHN Ngy 10 thng 11 nm 2014 1 / 28
http://find/http://goback/


2/31

Mc lc

1 Phng trnh tch bin

2 Phng trnh thun nht

3 Phng trnh vi phn ton phn

4 Phng trnh tuyn tnh cp mt

5 Phng trnh Bernoulli

6 Phng trnh vi phn tuyn tnh cp hai h s hng



3/31


4/31

Phng trnh tch bin

V d 1.2

Gii cc phng trnh sau:

1dydx

=6y2xviy(1) = 125

.

2 y =3x3 +4x 4

2y 4 viy(1) =3.

3 y = xy3

1+x2viy(0) = 1

4 y =ey(2x 4)viy(5) =0.

5 drd = r

2

vir(1) =2.

6dydt

=eyt1+t2

cos y viy(0) =0.

7 y

(4x

y+1)2 =0 viy(0) =2.

http://find/


5/31

Phng trnh thun nht

nh ngha 2.1

Hm s hai bin f(x, y)tha iu kin

f(k.x; k.y) =f(x, y),k=0.

Mi hm s thun nht u biu din c di dng

f(x, y) =f(1,y

x

) =g(y

x

) =g(u),

x=0

http://find/


6/31


7/31

Phng trnh thun nht

V d 2.1Kim tra cc phng trnh sau l phng trnh thun nht v giichng

1

y

=

x2

xy+y2

xy .

2dydx

=y+

x2 y2x

vix>0.

3 xyy +4x2 +y2 =0 viy(2) =7 vx>0. a

4 xy

=y(ln x ln y)viy(1) =4 vx>0.aNgun http://tutorial.math.lamar.edu/Classes/DE/Substitutions.aspx

http://find/


8/31

Phng trnh vi phn ton phn

nh ngha 3.1Phng trnh vi phn ton phn l phng trnh c dng

P(x, y)dx+Q(x, y)dy =0

trong P v Q l cc hm s hai bin x, y tha iu kin Py

=Qx

.

Cch gii 3.1

Tm hm (x, y)l th v ca trng vectorF =Pi +QjTch phn tng qutca phng trnh trn c dng(x, y) =C



9/31

Phng trnh vi phn ton phn

V d 3.1Kim tra c phng trnh sau l phng trnh vi phn ton phn. Giicc phng trnh .

1 (3y2 +2xy+2x)dx+ (6xy+x2 +3)dy =0.2 (x+y 1)dx+ (ey +x)dy=0.3 2xy 9x2 + (2y+x2 +1) dy

dx =0 viy(0) = 3.

4 4xy2 +4=2(3 x2y)y =0 viy(1) =8.5 2tyt2 +1 2t (2 ln(t2 +1))y

=0 viy(5) =0

6 3y3e3xy 1+ (2ye3xy +3xy2e3xy)y =0 viy(0) =1.

http://find/


10/31


11/31

Tha s tch phn

Bi ton 3.1Gi s phng trnh M(x, y)dx+N(x, y)dy =0 khng phi l phngtrnh vi phn ton phn. Hy tm mt hm s= (x, y)sao cho phngtrnh

(x, y).M(x, y)dx+ (x, y).N(x, y)dy =0l phng trnh vi phn ton phn.

nh ngha 3.2

Hm s(x, y)trong bi ton trn c gi l tha s tch phn caphng trnh cho.

http://find/


12/31

Tha s tch phn

V d 3.2

Xt phng trnh ydx+dy=0. Khi , hm s(x) =ex lm chophng trnh

exydx+exdy=0

l phng trnh vi phn ton phn. Do , (x) =ex l mt hm s cn

tm.

Cng thc 3.1

1 NuMy Nx

N

=f(x)-hm mt bin xth phng trnh c tha s

tch phn dng= (x)v (x) =e f(x)dx.

2 NuNxMy

M =f(y)-hm mt bin yth phng trnh c tha s

tch phn dng= (y)v (y) =e f(y)dy.

http://find/


13/31

Tha s tch phn

V d 3.3Bng cch tm tha s tch phn hy gii cc phng trnh vi phn sau

1 (2xy+x2y+y3

3)dx+ (x2 +y2)dy=0

2

y(1+xy)dx xdy=03 (x+y2)dx+xydy=04 (x2 +2y)dx xdy=05 (xex +xln y+y)dx+

x2

y

+xln y+xsin y dy=06 2y2(x+y2)dx+xy(x+6y2)dy=07 ydx (2x+y3ey)dy=0

http://find/


14/31

Phng trnh tuyn tnh cp mt

Phng trnh tuyn tnh cp mt l phng trnh c dng

y

+P(x).y=Q(x)

Cng thc nghim:y=eP(x)dx

Q(x)e

P(x)dxdx+C

http://find/


15/31

Phng trnh tuyn tnh cp mt

Phng trnh tuyn tnh cp mt l phng trnh c dng

y

+P(x).y=Q(x)

Cng thc nghim:y=eP(x)dx

Q(x)e

P(x)dxdx+C

http://find/


16/31

Ph h B lli


17/31

Phng trnh Bernoulli

nh ngha 5.1Phng trnh Bernoulli l phng trnh c dng

y +P(x).y=Q(x).y

vi Q(x)khng ng nht0v l hng s khc 0 v khc 1.

Cch gii 5.1

Xem xt y=0c l nghim ca phng trnh hay khng.

Gi s y=0. Chia hai v ca phng trnh cho y v t z=y1

Chuyn phng trnh v dng tuyn tnh cp mt vi n hm z.

Gii phng trnh ny v suy ra nghim y.


Ph h B lli
http://find/


18/31

Phng trnh Bernoulli

V d 5.1Xc nh gi tr trong cc phng trnh Bernoulli sau v gii chng

1 y +1x

y=xy2.

2 dydx.x3 sin y+2y=xdydx

3 y +4x

y=x3y2 viy(2) = 1vx>04 y =5y+e2xy2 viy(0) =2

5 6y 2y =xy4 viy(0) = 26 y +

yx

=

xviy(1) =0


Ph h h h i h h
http://find/


19/31

Phng trnh tuyn tnh cp hai h s hng

nh ngha 6.1

Phng trnh tuyn tnh cp hai h s hng l phng trnh c dng

y +py +qy=f(x) (1)

vi p, q l cc hng s.Nu f(x) 0th ta c phng trnhthun nht.Nu f(x)khng ng nht0th ta c phng trnhkhng thunnht.

http://find/


20/31


21/31



22/31


V d 6.1

Xc nh phng trnh c trng, sau vit nghim tng qut cacc phng trnh thun nht sau:

1 y +5y

6y=0.2 y 4y=0.3 y +2y =0.4 y +2y +y=0.5

y

+2y

+2y=0.6 y +9y=0.

http://find/


23/31

V phi c dng f (x) = ex P (x)


24/31

V phi c dng f(x) =e .Pn(x)

Nu . . .ca phng trnh c trng thYc dng . . .

khngl nghim Y =exQn(x)

lnghimn Y =exxQn(x)

l nghimkp Y =exx2Qn(x)

trong Qn(x)l a thc bcnm cc h s ca n cn c xc nh.


V phi c dng f (x) = ex P (x)
http://goforward/http://find/http://goback/


25/31

V phi c dng f(x) =e .Pn(x)

CC TRNG HP C BIT

Nu . . . v nu . . .ca p.trnh c trng thYc dng . . .

f(x) =a.ex khngl nghim Y =Aex

lnghimn Y =Aexxlnghimkp Y =Aexx2

f(x) =ax+b 0khngl nghim Y =Ax+B0l nghimn Y =x(Ax+B)0lnghimkp Y =x2(Ax+B)

f(x) =ax2 +bx+c 0khngl nghim Y =Ax2 +Bx+C0l nghimn Y =x(Ax2 +Bx+C)0lnghimkp Y =x2(Ax2 +Bx+C)

trong A,B,Cl cc h s cn c xc nh.CBGD. L Hoi Nhn () Chng 5. PHNG TRNH VI PHN Ngy 10 thng 11 nm 2014 22 / 28

Nghim ring ca phng trnh vi v phi
http://find/


26/31

Nghim ring ca phng trnh vi v phif(x) =ex.Pn(x)

V d 6.2

Xc nh tham s trong cc phng trnh sau y, nhn nh c l

nghim ca phng trnh c trng hay khng v gii cc phng trnh

1 y 3y +2y=4x.2 y 3y +2y=ex(3 4x)3 y

4y

+4y=4.e2x


http://find/


27/31

Nghim ring ca phng trnh vi v phif(x) =ex (Ur(x). cos x+Vs(x). sin x)

Nu . . .ca phng trnh c trng thYc dng . . .

ikhngl nghim Y =ex (Pn(x). cos x+Qn(x). sin x)

i lnghimn Y =exx(Pn(x). cos x+Qn(x). sin x)

Trong n=max(r, s);Pn(x)vQn(x)l a thc bcnm cc h sca chng cn c xc nh.




28/31


CC TRNG HP C BITNu . . . v nu . . .

ca p.trnh c trng thYc dng . . .

f(x) =a. sinx ikhngl nghim Y =A cos x+Bsin xi lnghimn Y =x(A cos x+Bsinx)

f(x) =a. cosx ikhngl nghim Y =A cos x+Bsin xi lnghimn Y =x(A cos x+Bsinx)


http://find/


29/31


V d 6.3

Gii phng trnh1 y +y 2y=cos x 3sin x.2 y +y=4x. sin x.


Nguyn l chng cht nghim
http://find/


30/31


nh l 6.1Nu Y1v Y2ln lt l nghim ca phng trnh:

y +py +qy=f1(x)

vy +py +qy=f2(x)

thY1+Y2l nghim ca phng trnh

y

+py

+qy=f1(x) +f2(x).




31/31


V d 6.4

Tmdng nghim tng qut ca cc phng trnh sau:

1 y +y=xex +2ex.2 y 2y =2 cos2 x.3 y 4y +4y=sin xsin 3x.


Documents

01 Slide - Chuong 4 Phuong Trinh Vi Phan