1
52.1
3.|/
Z?1 Vs ?M
Z?,i p3.
(2) dydx = x?y+1x+y?3:
(8) dydx = x2+y2xy,
(14) dydx = sin(x+y +1):
3:
(2)
7x =? +1,y = · +2, V|eZ?M1
d·
d? =
·
+·;
7u = ·?,5μ
dud? +u = 1?u1+u;
¨s ?M
E p¤ Y31u2 +2u?1 = C12, ?C11 ?iè
.?
u = ·? = y?2x?1;? = x?1;
} ?

Tie¤eZ?￥Y31
y2 +2xy?x2?6y?2x = C;
?C1 ?iè
.
(8)|eZ??1
dy
dx =
x
y +
y
x;
7u = yx,5μ
xdudx +u = 1u +u;
¨s ?M
E p¤ Y31u2 = lnx2 +C,|uD?yx¤eZ?￥Y31y2 = x2(lnx2 +C),
?C1 ?iè
.
(14)
7u = x+y +1,5μ
du
dx = 1+sinu;
¨s ?M
E p¤ Y31tanu?secu = x+C,|uD?x+y +1¤eZ?￥Y31
tan(x+y +1)?sec(x+y +1) = x+C;
?C1 ?iè
.
4.3/
L?±sZ?.
(1) dydx? 2yx+1 = (x+1)52:
(6) dydx?2xy = x:
3:
2
(1)? úa(x) = 2x+1,f(x) = (x+1)52.V7 V peZ?￥Y31
y = exp(
Z 2
x+1dx)(C +Z
(x+1)52 exp(?
Z 2
x+1dx)dx)
= C(x+1)2 + 23(x+1)72;
'y = C(x+1)2 + 23(x+1)72, ?C1 ?iè
.
(6)? úa(x) = 2x,f(x) = x.V7 V peZ?￥Y31
y = exp(2
Z
xdx)(C +
Z
xexp(?2
Z
xdx)dx)
=?12 +Cex2;
'y =?12 +Cex2, ?C1 ?iè
.
6. p/
′ù5￥3.
(2) y(1+x2)dy = x(1+y2)dx; y(0) = 1.
(5) ey dydx?x?x3 = 0; y(1) = 1.
3:
(2)?
^M
s ?￥Z?,s ?M
a¤
y
1+y2dy =
x
1+x2dx;

s¤ Y311+y2 = C(1+x2), ?C1 ?iè
.} ?′Hq¤C = 2.#
ó
′ù5￥31y = p1+2x2.
(5)
7u = ey,eZ?M1
du
dx = x+x
3;
?^ p¤ Y31u = 12x2 + 14x4 +C,V7eZ?￥Y31
ey = 12x2 + 14x4 +C;
?C1 ?iè
.} ?′Hq¤e = 34 +C,V7C = e? 34.#
ó′ù5￥31
ey = 12x2 + 14x4 +e? 34:
7. p3/
BernoulliZ?
(1) dydx = 6yx?xy2:
(3) xdydx?4y = 2x2py (x 6= 0;y > 0):
3:
(1)?y 6= 0
H,
7z = y?1,eZ?M1
dz
dx =?
6
xz +x;
?
^B¨L?±sZ?, Y31
z = 1x6(C + 18x8);
3
V7eZ?￥Y31
x6
y?
x8
8 = C;
?C1 ?iè
.N?,A ?y = 09
^Z?￥3.
(3)
7z = py,eZ?M1
dz
dx =
2
xz +x;
?
^B¨L?±sZ?, Y31z = x2(lnjxj+C),V7eZ?￥Y31y = x4(lnjxj+C)2,
?C1 ?iè
.
11.
!y1(x),y2(x)
^Z?
dy
dx +p(x)y = q(x);
￥
?os3. p￡??Z?￥ ?B3y(x),? ??
T
y(x)?y1(x)
y2(x)?y1(x) = C;
?C
^
è
.
￡
ü:
7?(x) = y(x)?y1(x),`(x) = y2(x)?y1(x), ?^￡
d?
dx +p(x)?(x) = 0;
d`
dx +p(x)`(x) = 0:
yNiè
k1,k2 6= 0
P¤
(x) = k1 exp(?
Z x
x0
p(t)dt); `(x) = k2 exp(?
Z x
x0
p(t)dt);
?x0 2R.V7
y(x)?y1(x)
y2(x)?y1(x) =
(x)
`(x) =
k1
k2 = C;
? úC = k1k21Bè
.