1
B`
AI T ˆ
A
.P PHU
.O
.NG TR`
INH VI PH ˆ
AN
1)
Gia
'i phu.o.ng trnh:
2xy0y” = y02−1
HD gia
’i:
D
-a
.t
y0=p: 2xpp0=p2−1
Vo.i
x(p2−1) 6= 0
ta co:
2pdp
p2−1=dx
x⇔p2−1 = C1⇔p=±√C1x+ 1
p=dy
dx =√C1+ 1 ⇒y=2
3C1
(C1x+ 1)3
2+C2
2)
Gia
'i phu.o.ng trnh:
√y.y” = y0
HD gia
’i:
D
-a
.t
y0=p⇒y” = pdp
dy
(ham theo y). Phu.o.ng trnh tro.
'thanh:
√ypdp
dy =p
Vo.i
p6= 0
ta du.o.
.c phu.o.ng trnh:
dp =dy
√y⇒p= 2√y+C1⇔dy
dx = 2√y+C1⇒
dx =dy
2√y+C1
Tu.do nghi^e
.m t^o
'ng quat:
x=√y−C1
2ln |2√y+C1|+C2
Ngoai ra
y=c
: ha
ng cu
~ng la nghi^e
.m.
3)
Gia
'i phu.o.ng trnh:
a(xy0+ 2y) = xyy0
HD gia
’i: a(xy0+ 2y) = xyy0⇒x(a−y)y0=−2ay
N^e
u
y6= 0
, ta co phu.o.ng trnh tu.o.ng du.o.ng vo.i
a−y
ydy =−2a
xdx ⇔x2ayae−y=C
Ngoai ra
y= 0
cu
~ng la nghi^e
.m.
4)
Gia
'i phu.o.ng trnh:
y” = y0ey
HD gia
’i:
D
-a
.t
y0=p⇒y” = pdp
dy
thay vao phu.o.ng trnh:
pdp
dy =pey
Vo.i
p6= 0 : dp
dy =ey⇔p=ey+C1⇒dy
dx =ey+C1⇔dy
ey+C1
=dx
Vo.i
C16= 0
ta co:
Rdy
ey+C1
=1
C1Rey+C1−ey
ey+ 1 dy =1
C1
(y−Reydy
ey+C1
) = y
C1−
1
C1
ln(ey+C1)
nhu.v^a
.y:
Rdx
ey+C1
=
−e−ynˆe
´uC1= 0
1
C1
(y−ln |ey+C1|) nˆe
´uC16= 0.
Ngoai ra
y=C:
ha
ng la m^o
.t nghi^e
.m
5)
Gia
'i phu.o.ng trnh:
xy0=y(1 + ln y−ln x)
vo.i
y(1) = e
.
2
HD gia
’i:
D
-u.a phu.o.ng trnh v^e
:
y0=y
x(1 + ln y
x)
, da
.t
y=zx
du.o.
.c:
xz0=zln z
•zln z6= 0 ⇒dz
zln z=dx
x⇒ln z=Cx
hay
ln y
x=Cx ⇔y=xeCx
y(1) = e→C= 1.
V^a
.y
y=xex
6)
Gia
'i phu.o.ng trnh:
y”(1 + y) = y02+y0
HD gia
’i:
D
-a
.t
y0=z(y)⇒z0=zdz
dy
thay vao phu.o.ng trnh:
dz
z+ 1 =dy
y+ 1
⇒z+ 1 = C1(y+ 1) ⇒z=C1y+C1−1⇔dy
C1y+C1−1=dx (∗)
•C1= 0 ⇒(∗)
cho
y=C−x
•C16= 0 ⇒(∗)
cho
1
C1
ln |C1y+C1−1|=x+C2
Ngoai ra
y=C
la nghi^e
.m.
Tom la
.i nghi^e
.m t^o
'ng quat:
y=C, y =C−x;1
C1
ln |C1y+C1−1|=x+C2
7)
Gia
'i phu.o.ng trnh:
y0=y2−2
x2
HD gia
’i:
Bi^e
n d^o
'i (3) v^e
da
.ng:
x2y0= (xy)2−2 (∗)
D
-a
.t
z=xy ⇒z0=y+xy0
thay vao
(∗)
suy ra:
xz0=z2+z−2⇔dz
z2+z−2=dx
x⇔3
rz−1
z+x=Cx
V^a
.y TPTQ:
xy −1
xy + 2 =Cx3.
8)
Gia
'i phu.o.ng trnh:
yy” + y02= 1
HD gia
’i:
D
-a
.t
y0=z(y)⇒y” = z.dz
dy
Bi^e
n d^o
'i phu.o.ng trnh v^e
:
z
1−z2dz =dy
y⇔z2= 1 + C1
y2
⇒dy
dx =±r1 + C1
y2⇔ ±Rdy
r1 + C1
y2
=dx ⇒y2+C1= (x+C2)2
Nghi^e
.m t^o
'ng quat:
y2+C1= (x+C2)2
9)
Gia
'i phu.o.ng trnh:
2x(1 + x)y0−(3x+ 4)y+ 2x√1 + x= 0
HD gia
’i: y0−3x+ 4
2x(x+ 1).y =−1
√x+ 1;x6= 0, x 6=−1
Nghi^e
.m t^o
'ng quat cu
'a phu.o.ng trnh thu^a
n nh^a
t:
Rdy
y=R3x+ 4
2x(x+ 1)dx =R(2
x−1
2(x+ 1))dx ⇔y=Cx2
√x+ 1
.
3
Bi^e
n thi^en ha
ng s^o
:
C0=−1
x2⇒C=−1
x+ε.
V^a
.ey nghi^
.m t^o
'ng quat:
y=x2
√x+ 1(1
x+ε)
10)
Gia
'i phu.o.ng trnh:
y” = e2y
thoa
'
(y(0) = 0
y0(0) = 0
HD gia
’i:
D
-a
.t
z=y0→y” = z.dz
dy
phu.o.ng trnh tro.
'thanh
z.dz
dy =e2y⇔z2
2=e2y
2+ε
y0(0) = y(0) = 0 ⇒ε=−1
2.
V^a
.y
z2=e2y−1.
Tu.do:
z=dy
dx =√e2y−1⇒Zdy
√e2y−1=x+ε. d¯ˆo
’i biˆe
´nt=√e2y−1
arctg√e2y−1 = x+ε
y(0) = 0 ⇒ε= 0.
V^a
.y nghi^e
.m ri^eng thoa
'di^e
u ki^e
.n d^e
bai:
y=1
2ln(tg2x+ 1).
11)
Tm nghi^e
.m ri^eng cu
'a phu.o.ng trnh:
xy0+ 2y=xyy0
thoa
'ma
~n di^e
u ki^e
.n d^a
u
y(−1) = 1
.
HD gia
’i:
Vi^e
t phu.o.ng trnh la
.i:
x(1 −y)y0=−2y
; do
y(−1) = 1
n^en
y6≡ 0
. D
-u.a v^e
phu.o.ng trnh tach bi^e
n:
1−y
ydy =−2dx
x
tch ph^an t^o
'ng quat:
x2ye−y=C
. Thay di^e
u ki^e
.n vao ta du.o.
.c
C=1
e
. V^a
.y tch ph^an
ri^eng c^a
n tm la:
x2ye1−y= 1
.
12)
Ba
ng cach da
.t
y=ux
, ha
~y gia
'i phu.o.ng trnh:
xdy −ydx −px2−y2dx = 0.(x > 0)
HD gia
’i:
D
-a
.t
y=ux;du =udx +xdu
thay vao phu.o.ng trnh va gia
'n u.o.c
x
:
xdu −
√1−u2dx = 0
. Ro
~rang
u−±1
la nghi^e
.m. khi
u6≡ ±1
du.a phu.o.ng trnh v^e
tach bi^e
n:
du
1−u2=dx
x
. TPTQ:
arcsin u−ln x=C
(do
x > 0
).
V^a
.y NTQ cu
'a phu.o.ng trnh:
y=±x; arcsin y
x= ln x+C
.
13)
Tm nghi^e
.m ri^eng cu
'a phu.o.ng trnh:
xy0=px2−y2+y
thoa
'ma
~n di^e
u ki^e
.n d^a
u
y(1) = 0
.
HD gia
’i:
xy0=px2−y2+y⇐⇒ y0=r1−y2
x2+y
x
da
.t
u=y
x
hay
y=ux
suy ra
y0=xu0+u
phu.o.ng trnh thanh:
xu0=√1−u2⇐⇒ du
√1−u2=dx
x
.
4
⇐⇒ arcsin u= ln Cx
thoa
'ma
~n di^e
u ki^e
.n d^a
u
y(1) = 0
khi
C= 1
. V^a
.ey nghi^
.m
y=±x
.
14)
Tm nghi^e
.m ri^eng cu
'a phu.o.ng trnh:
y0sin x=yln y
thoa
'ma
~n di^e
u ki^e
.n d^a
u
y(π
2) = e
.
HD gia
’i:
y0sin x=yln y⇐⇒ dy
yln y=dx
sin x
⇐⇒ ln y=Ctan x
2⇐⇒ y=eCtan x
2
thoa
'ma
~n di^e
u ki^e
.n d^a
u
y(π
2) = e
khi
C= 1
. V^a
.y
y=etan x
2
.
15)
Tm nghi^e
.m ri^eng cu
'a phu.o.ng trnh:
(x+y+ 1)dx + (2x+ 2y−1)dy = 0
thoa
'ma
~n di^e
u ki^e
.n d^a
u
y(0) = 1
.
HD gia
’i:
D
-a
.t
x+y=z=⇒dy =dz −dx
phu.o.ng trnh thanh:
(2 −z)dx + (2z−1)dz = 0
; gia
'i ra
x−2z−3 ln |z−2|=C
. V^a
.y
x+ 2y+ 3 ln |x+y−2|=C
thoa
'ma
~n di^e
u ki^e
.n d^a
u
y(0) = 1
khi
C= 2
.
16)
Ba
ng cach da
.t
y=1
z
r^o
ai d
.t
z=ux
,ha
~y gia
'i
phu.o.ng trnh:
(x2y2−1)dy + 2xy3dx = 0
HD gia
’i:
D
-a
.t
y=1
z
du.o.
.c:
(z2−x2)dz + 2zxdx = 0
r^o; ai d
.t
z=ux
du, .o.
.c
(u2−1)(udx +xdu)+2udx = 0
⇐⇒ dx
x+u2−1
u3+udu = 0
⇐⇒ ln |x|+ ln u2+ 1
|u|= ln C⇐⇒ x(u2+ 1)
u=C
thay
u=1
xy
du.o.
.c nghi^e
.m
1 + x2y2=Cy
.
17)
eTm nghi^
.m t^o
'ng quat cu
'a phu.o.ng trnh sau:
y0−xy =x+x3
HD gia
’i:
D
-^ay la phu.o.ng trnh tuy^e
n tnh c^a
ep 1 va co nghi^
.m t^o
'ng quat la
y=Cex2
2.x2
2+ 1
.
.
5
18)
eTm nghi^
.m t^o
'ng quat cu
'a cac phu.o.ng trnh sau:
y0−y=y2.
HD gia
’i:
D
-^ay la phu.o.ng trnh tach bi^e
en va co nghi^
.m t^o
'ng quat la
ln |y
y+ 1|=x+C.
19)
Tm nghi^e
.m cu
'a cac phu.o.ng trnh sau:
y0+y
x=ex
HD gia
’i:
D
-^ay la phu.o.ng trnh tuy^e
n tnh c^a
ep 1 va co nghi^
.m t^o
'ng quat la
y=C
x+ex−ex
x
.
20)
Tm nghi^e
.m cu
'a cac phu.o.ng trnh sau:
y0−y=y3.
HD gia
’i:
D
-^ay la phu.o.ng trnh tach bi^e
en va co nghi^
.m t^o
'ng quat la
C+x= ln |y| − arctgy.
21)
Gia
'i phu.o.ng trnh:
y0=y
x+ sin y
x
, vo.i
y(1) = π
2
HD gia
’i: y=zx ⇒y0=z0x+z
, phu.o.ng trnh tro.
'thanh:
z0x= sin x⇔dz
sin z=dx
x⇔ln |tg z
2|= ln |x|+ ln C⇔tg z
2=Cx
V^a
.ey nghi^
.m t^o
'ng quat:
tg y
2x=Cx;y(1) = π
2⇒C= 1.
V^a
.y:
tg y
2x=x
.
22)
Gia
'i phu.o.ng trnh:
(x−ycos y
x)dx +xcos y
xdy = 0
HD gia
’i:
D
-a
.t
y
x=z⇒y0=z0x+z
phu.o.ng trnh du.o.
.c du.a v^e
da
.ng:
xcos z.z0+ 1 = 0 ⇔Zcos zdz =−dx
x+C⇔sin z=−ln |x|+C
V^a
.y TPTQ:
sin y
x=−ln |x|+C
23)
Gia
'i phu.o.ng trnh:
(y02−1)x2y2+y0(x4−y4) = 0
HD gia
’i:
La phu.o.ng trnh da
'ng c^a
p nhu.ng gia
'i kha phu.c ta
.p.
.
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