chuong3_phuongtrinhviphancap2

Chng 3. Phng trnh vi phncp 2

ThS. Cao Xun Phng

01/01/2014 - C01004

Cao Xun Phng (Khoa Ton-Thng k) Ton cao cp A4 01/01/2014 - C01004 1 / 21

Ni dung

1 Cc khi nim

2 PTVP cp 2 khuyt y, y

3 PTVP cp 2 khuyt y

4 PTVP cp 2 khuyt x

5 PTVP tuyn tnh cp 2 thun nht

6 PTVP tuyn tnh cp 2 khng thun nht


Cc khi nim

nh ngha

Phng trnh vi phn cp 2 l phng trnh hm dng

F (x , y (x) , y (x) , y (x)) = 0, (1)

trong

x l bin c lp;

y(x) l gi tr ca hm y ti x ;

y (x) l gi tr ca o hm y ti x ;y (x) l gi tr ca o hm cp hai y ti x .


Cc khi nim

nh ngha

Nghim ca PTVP (1) l mt hm s y = y0(x)

xc nh v c o hm cho n cp 2 trn mtkhong I R no sao cho khi ta thay n vo(1) th ta nhn c mt ng thc ng.

Gii mt PTVP cp 2 dng (1) l ta i tm tp hpnghim ca n.


PTVP cp 2 khuyt y, y

Dng.y = f (x). (2)

Cch gii. Ly tch phn hai v ca (2), ta c

y =

f (x)dx g(x) + C1. (3)

Tip tc ly tch phn hai v ca (3), ta c

y =

[g(x) + C1] dx h(x) + C1x + C2. (4)


PTVP cp 2 khuyt y, y

V d 1. Gii cc phng trnh vi phn sau:

a) y = e2x , y(0) = 74, y (0) =

3

2.

b) y cos2 x 1 = 0.


PTVP cp 2 khuyt y

Dng.y = f (x , y ). (5)

Cch gii. t z = y . Ta c phng trnh

z = f (x , z) . (6)

Ta thy (6) l mt PTVP cp 1. Nu gii c, ta c

nghim tng qut l

z = z (x ,C1) .

Khi , nghim tng qut ca (5) l

y =

z (x ,C1) dx .


PTVP cp 2 khuyt y


a) y = 3x y

x, (x > 0).

b)(1+ x2

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


PTVP cp 2 khuyt x

Dng.y = f (y , y ) . (7)

Cch gii.

t y = p = p (y) v xem nh l hm ca y . Lyo hm hai v ca hm ny theo x , ta c

y =dp

dx=

dp

dy.dy

dx=

dp

dy.y = p

dp

dy. (8)

Khi , (7) tr thnh

pdp

dy= f (y , p) . (9)


PTVP cp 2 khuyt x

Ta xem (9) l PTVP cp 1 vi n hm p. Nu giic, ta c nghim p = p (y ,C1).

Cui cng, t quan h

dy

dx= y = p = p (y ,C1) ,

ta nhn c

x =

dy

p (y ,C1).


PTVP cp 2 khuyt x


a) y = y ey .

b) (y )2 + yy = 0, y(1) = 2, y (1) =1

2.


PTVP tuyn tnh cp 2 thun nht

Dng.ay + by + cy = 0, (a 6= 0). (10)

Cch gii. Xt phng trnh c trng

ak2 + bk + c = 0. (11)

Ta c cc trng hp sau:

Nu (11) c hai nghim thc phn bit k1, k2, th

nghim tng qut ca (10) c dng

y = C1ek1x + C2e

k2x .



Nu (11) c nghim kp k1 = k2 = k , th nghim

tng qut ca (10) c dng

y = C1ekx + C2xe

kx .

Nu (11) c hai nghim phc lin hp k1 = + i ,

k2 = i , th nghim tng qut ca (10) c dngy = ex (C1 cos x + C2 sin x) .




a) y 7y + 12y = 0.b) 4y + 12y + 9y = 0.c) y 2y + 5 = 0.d) y 4y + 3y = 0, y(0) = 6, y (0) = 10.


PTVPTT cp 2 khng thun nht

Dng.

ay + by + cy = f (x), (a 6= 0). (12)Cch gii.

Tm nghim tng qut ytn ca phng trnh thunnht tng ng, ay + by + cy = 0.Tm mt nghim ring yr ca phng trnh khngthun nht (12).

Kt lun: Nghim tng qut ca (12) l

yktn = ytn + yr .



V d 5. Cho phng trnh vi phn

y 2y + 2y = (x2 + 2) ex .a) Chng minh rng hm y = x2ex l mt nghim caphng trnh trn.

b) Gii phng trnh cho.



Tm nghim ring ca phng trnh khng thunnht (12) trong mt s trng hp c bit

Trng hp 1: V phi f (x) = exPn (x), vi Pn(x) la thc bc n theo bin x .

Khi khng phi l nghim ca phng trnh ctrng ca phng trnh thun nht tng ng, tatm nghim ring yr di dng

yr = exQn (x) .



Khi l nghim n ca phng trnh c trngca phng trnh thun nht tng ng, ta tmnghim ring yr di dng

yr = xexQn (x) .

Khi l nghim kp ca phng trnh c trngca phng trnh thun nht tng ng, ta tmnghim ring yr di dng

yr = x2exQn (x) .




a) y 5y + 4y = 4x2.b) y 3y + 2y = (2x + 1) ex .c) y 6y + 9y = x + 1.



Trng hp 2: V phi dng

f (x) = ex [Pn (x) cos x + Qm (x) sin x ] ,

vi Pn(x), Qm(x) ln lt l cc a thc c bc n, m.

Khi i khng phi l nghim ca phng trnhc trng, ta tm nghim ring yr di dng

yr = ex [Hk (x) cos x + Rk (x) sin x ]

vi k = max {m, n}.



Khi i l nghim ca phng trnh c trng,ta tm nghim ring yr di dng

yr = xex [Hk (x) cos x + Rk (x) sin x ]

vi k = max {m, n}.




a) y 3y + 2y = cos x .b) y 2y = ex sin x .

KT THC CHNG


Cc khi nimPTVP cp 2 khuyt y, y'PTVP cp 2 khuyt yPTVP cp 2 khuyt xPTVP tuyn tnh cp 2 thun nhtPTVP tuyn tnh cp 2 khng thun nht

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