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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
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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
CBGD. L Hoi Nhn () Chng 5. PHNG TRNH VI PHN Ngy 10 thng 11 nm 2014 2 / 28
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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.
CBGD. L Hoi Nhn () Chng 5. PHNG TRNH VI PHN Ngy 10 thng 11 nm 2014 4 / 28
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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
CBGD. L Hoi Nhn () Chng 5. PHNG TRNH VI PHN Ngy 10 thng 11 nm 2014 5 / 28
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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
CBGD. L Hoi Nhn () Chng 5. PHNG TRNH VI PHN Ngy 10 thng 11 nm 2014 7 / 28
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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
CBGD. L Hoi Nhn () Chng 5. PHNG TRNH VI PHN Ngy 10 thng 11 nm 2014 8 / 28
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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.
CBGD. L Hoi Nhn () Chng 5. PHNG TRNH VI PHN Ngy 10 thng 11 nm 2014 9 / 28
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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.
CBGD. L Hoi Nhn () Chng 5. PHNG TRNH VI PHN Ngy 10 thng 11 nm 2014 10 / 28
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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.
CBGD. L Hoi Nhn () Chng 5. PHNG TRNH VI PHN Ngy 10 thng 11 nm 2014 11 / 28
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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
CBGD. L Hoi Nhn () Chng 5. PHNG TRNH VI PHN Ngy 10 thng 11 nm 2014 12 / 28
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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
CBGD. L Hoi Nhn () Chng 5. PHNG TRNH VI PHN Ngy 10 thng 11 nm 2014 13 / 28
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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
CBGD. L Hoi Nhn () Chng 5. PHNG TRNH VI PHN Ngy 10 thng 11 nm 2014 13 / 28
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Ph h B lli
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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.
CBGD. L Hoi Nhn () Chng 5. PHNG TRNH VI PHN Ngy 10 thng 11 nm 2014 15 / 28
Ph h B lli
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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
CBGD. L Hoi Nhn () Chng 5. PHNG TRNH VI PHN Ngy 10 thng 11 nm 2014 16 / 28
Ph h h h i h h
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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.
CBGD. L Hoi Nhn () Chng 5. PHNG TRNH VI PHN Ngy 10 thng 11 nm 2014 17 / 28
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Phng trnh tuyn tnh cp hai h s hng
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Phng trnh tuyn tnh cp hai h s hng
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.
CBGD. L Hoi Nhn () Chng 5. PHNG TRNH VI PHN Ngy 10 thng 11 nm 2014 19 / 28
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V phi c dng f (x) = ex P (x)
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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.
CBGD. L Hoi Nhn () Chng 5. PHNG TRNH VI PHN Ngy 10 thng 11 nm 2014 21 / 28
V phi c dng f (x) = ex P (x)
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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
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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
CBGD. L Hoi Nhn () Chng 5. PHNG TRNH VI PHN Ngy 10 thng 11 nm 2014 23 / 28
Nghim ring ca phng trnh vi v phi
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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.
CBGD. L Hoi Nhn () Chng 5. PHNG TRNH VI PHN Ngy 10 thng 11 nm 2014 24 / 28
Nghim ring ca phng trnh vi v phi
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Nghim ring ca phng trnh vi v phif(x) =ex (Ur(x). cos x+Vs(x). sin x)
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)
CBGD. L Hoi Nhn () Chng 5. PHNG TRNH VI PHN Ngy 10 thng 11 nm 2014 25 / 28
Nghim ring ca phng trnh vi v phi
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Nghim ring ca phng trnh vi v phif(x) =ex (Ur(x). cos x+Vs(x). sin x)
V d 6.3
Gii phng trnh1 y +y 2y=cos x 3sin x.2 y +y=4x. sin x.
CBGD. L Hoi Nhn () Chng 5. PHNG TRNH VI PHN Ngy 10 thng 11 nm 2014 26 / 28
Nguyn l chng cht nghim
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Nguyn l chng cht nghim
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).
CBGD. L Hoi Nhn () Chng 5. PHNG TRNH VI PHN Ngy 10 thng 11 nm 2014 27 / 28
Nguyn l chng cht nghim
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Nguyn l chng cht nghim
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.
CBGD. L Hoi Nhn () Chng 5. PHNG TRNH VI PHN Ngy 10 thng 11 nm 2014 28 / 28
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