1
52.4
1. p/
Z?￥Y3.
(2) y = (dydx)2?xdydx + x22,
(3) y2(1? dydx) = (2? dydx)2:
(4) (dydx)3?4xydydx +8y2 = 0:
3:
(2)
7p = dydx,5eZ?M1:
y = p2?xp+ x
2
2,


Z?
H1?x p?,¤
p = 2pdpdx?p?xdpdx +x;
'
(2p?x)dpdx = (2p?x);
?2p? x 6= 0,5dpdx = 1,V7p = x + C, ?C1 ?iè
,y7eZ?￥Y31
y = x22 +Cx+C2. ?2p?x = 0, ?^ peZ?￥
6B3y = x24,
(3)
7dydx = p,2?p = yt,?Z? V¤
p = 1?t2; y = t+ 1t:
?p 6= 0
H,¤:
dx = dyp =? 1t2dt:
5
x =?
Z 1
t2dt =
1
t +C:
yN,eZ?￥?
?
T￥31
8
><
>:
x = 1t +C;
y = t+ 1t;
?C1 ?iè
.h ??
a¤:
y = x+ 1x?C?C:
N?,?p = 0
H,^?y = §29
^Z?￥3.
(4)
7p = dydx,5
x = p
2
4y +
2y
p,


Z?
H1?x p?,¤
1 = ( p2y? 2yp2 )pdpdy? p
2
4y2 +2;
2
'
p3?4y2
2yp2
dp
dy =
p3?4y2
4y2p ;
?N¤
dp
dy =
p
2yp
3?4y2 = 0:
?dpdy = p2y¤p = C1y12, ?C11 ?iè
.#
x = C
2
1
4 +
2y12
C1 ;
'y = C(x?C)2, ?C = C214,
?p3?4y2 = 0¤p = (4y2)13,?N¤Z?￥
6B?3
x3 = 274 y;
'y = 427x3.
2.3/
Z?,i p 3( ?Ti￥).
(1) (dydx)2 +y2?1 = 0:
(2) x(dydx)2?ydydx +1 = 0:
(6) dydx =?x+px2 +2y:
3:
(1)
7dydx = p,?Z? V¤
p = cost; y = sint:
?p 6= 0
H,¤:
dx = dyp = dt:
5x = t?C.yN,eZ?￥?
?
T￥31
x = t?C;
y = sint;
?C1 ?iè
.h ??
a¤y = sin(x + C).N?,?p = 0
H,^?y = §19
^Z?
￥3.
s wLBy = sin(x+C)￥C -
Y wL
@Z?:
y?sin(x+C) = 0;
cos(x+C) = 0:
V?h ?C¤y = §1,^￡
^eZ?￥ 3.
(2)
7p = dydx,?Z??p 6= 0.yN V3
y = xp+ 1p:


Z?
H1?x p?,¤
p = p+xdpdx? 1p2 dpdx;
3
'
(x? 1p2)dpdx = 0;
?x?p?2 6= 0,5dpdx = 0,V7p = C, ?C1 ?iè
,y7eZ?￥Y31y = Cx+ 1C.
?x?p?2 = 0, ?^ peZ?￥
6B3y2 = 4x.
s wLBy = Cx+ 1C￥C -
Y wL
@Z?:
y?Cx? 1
C = 0;?x+ 1
C2 = 0:
V?h ?C¤y2 = 4x,^￡
^eZ?￥ 3.
(6)
7z = x2 +2y,5?Z?¤
dz
dx = 2x+2
dy
dx = 2
pz:
?z 6= 0
H,¨s ?M
￥ZE¤z = (x+C)2,?N¤eZ?￥Y31y = Cx+ C22, ?C
1 ?iè
. ?z = 0, ?^ peZ?￥
6B3y =?x22,
s wLBy = Cx+ C22￥C -
Y wL
@Z?:
y?Cx? C2
2 = 0;?x?C = 0:
V?h ?C¤y =?x22,^￡
^eZ?￥ 3.
3. p/
ú¨Z?￥3.
(1) d5ydx5? 1x d4ydx4 = 0:
(10) 4d4ydx4 = d2ydx2:
3:
(1)
7p = d4ydx4,5eZ?M1
dp
dx =
p
x:
P¨s ?M
￥ZE, V¤p = A1x, ?A11 ?iè
.#
d4y
dx4 = A1x:
3¤y = C1x5 +C2x3 +C3x2 +C4x+C5, ?C1 = A15!,C2,C3,C4,C51 ?iè
.
(10)
7p = d2ydx2,5eZ?M1
4d
2p
dx2 = p;
[p0dxe

T

¤2d(p0)2 = pdp,#4(p0)2 = p2 +A1, ?A11 ?iè
.V7
2dpdx = §
p
p2 +A1:
I
nZ?
2dpdx =
p
p2 +A1;
P¨s ?M
￥ZE, V¤
p+
p
p2 +A1 = A2ex2 ;
4
?A21 ?iè
.?N V¤
p+
p
p2 +A1 = A2ex2 ;
yN
p = A22 ex2? A12A
2
e?x2 ;
'
d2y
dx2 =
A2
2 e
x
2? A12A
2
e?x2 ;
üV
Qsa¤eZ?￥Y3
y = C1ex2 +C2e?x2 +C3x+C4;
?C1 = 2A2,C2 =?2A1A2,C3,C41 ?iè
.
?Z?
2dpdx =?
p
p2 +A1;
V¤eZ?M]?
T￥Y3.
7.
k￡ ?y = ’(x)
^Z?
dy
dx = p(x)siny
￥
@
SHq’(0) = 0￥3,5’(x) · 0, ?p(x)?1 < x < 1
 ??.
￡
ü:¨Q￡E. ?’(x) 6· 0,5μx0
P¤’(x0) 6= 0,?^
!x0 > 0.
7
ˉx0 = supfx 2Rj’(x) = 0 Ox < x0g;
5A ?μ’(ˉx0) = 0 O?x 2 (ˉx0;x0]
H,’(x) 6= 0. uWx 2 (ˉx0;x0]
??’(x) 6= 0,yN?s
?M
E V?id
,è
K
P¤
tan y2 = K exp(
Z x
0
p(s)ds):


T

7x ! ˉx0,5P
t?0,7·
t?d
,è
K,?ü?á
±.#Aμ’(x) · 0.