1
52.2
1.￡/
Z?
^ ?Z?,i pZ?￥3:
(2) (cosx+ 1y)dx+(1y? xy2)dy = 0:
(3) (5x4 +3xy2?y3)dx+(3x2y?3xy2 +y2)dy = 0:
(5) dydx =?6x+y+2x+8y?3:
(9) 3y +ex +(3x+cosy)dydx = 0:
3:
(2)? úM(x;y) = cosx+ 1y,N(x;y) = 1y? xy2,??
@M
@y =?
1
y2 =
@N
@x ;
[?
^B? ?Z?. |x0 = 0,y0 = 1, V9

U(x;y) =
Z x
0
(cosx+ 1y)dx+
Z y
1
1
ydy
= sinx+ xy +lnjyj:
#?Z?￥Y31sinx+ xy +lnjyj = C, ?C1 ?iè
.
(3)? úM(x;y) = 5x4 +3xy2?y3,N(x;y) = 3x2y?3xy2 +y2,??
@M
@y = 6xy?3y
2 = @N
@x ;
[?
^B? ?Z?. |x0 = 0,y0 = 0, V9

U(x;y) =
Z x
0
(5x4 +3xy2?y3)dx+
Z y
0
y2dy
= x5 + 32x2y2?xy3 + y
3
3,
#?Z?￥Y31x5 + 32x2y2?xy3 + y33 = C, ?C1 ?iè
.
(5)|eZ??1
(6x+y +2)dx+(x+8y?3)dy = 0;
? úM(x;y) = 6x+y +2,N(x;y) = x+8y?3,??
@M
@y = 1 =
@N
@x ;
[?
^B? ?Z?. |x0 = 0,y0 = 0, V9

U(x;y) =
Z x
0
(6x+y +2)dx+
Z y
0
(8y?3)dy
= 3x2 +xy +2x+4y2?3y:
#?Z?￥Y313x2 +xy +2x+4y2?3y = C, ?C1 ?iè
.
2
(9)|eZ??1
(3y +ex)dx+(3x+cosy)dy = 0;
? úM(x;y) = 3y +ex,N(x;y) = 3x+cosy,??
@M
@y = 3 =
@N
@x ;
[?
^B? ?Z?. |x0 = 0,y0 = 0, V9

U(x;y) =
Z x
0
(3y +ex)dx+
Z y
0
cosydy
= ex +3xy +siny:
#?Z?￥Y31ex +3xy +siny = C, ?C1 ?iè
.
3.
k¨sy0E3/
Z?:
(1) ydx+(y?x)dy = 0:
(3) (x2 +y2 +y)dx?xdy = 0:
(5) 2xylnydx+(x2 +y2p1+y2)dy = 0:
3:
(1)? úM(x;y) = y,N(x;y) = y?x,??
E = @M@y? @N@x = 2;
[
?
^ ?Z?.?EM =?2yDxí1,yN?Z?μoG ??y￥sy0
(x) = e?
R 2
y dy =? 1y2:
yNZ?
1
ydx+
y?x
y2 dy = 0
1 ?Z?, |x0 = 0,y0 = 1, V9

U(x;y) =
Z x
0
1
ydx+
Z y
1
1
ydy = lnjyj+
x
y:
#?Z?￥Y31lnjyj+ xy = C, ?C1 ?iè
.N?,A ?y = 09
^Z?￥3.
(3)? úM(x;y) = x2 +y2 +y,N(x;y) =?x,??
E = @M@y? @N@x = 2(y +1);
[
?
^ ?Z?.YV43^?
(x2 +y2 +y)dx?xdy
= (x2 +y2)(dx+ ydx?xdyx2 +y2 )
= (x2 +y2)(dx+d(arctan(xy)))
= (x2 +y2)d(x+arctan(xy))
yN?Z?μsy01x2+y2, O Y31
x+arctan(xy) = C;
?C1 ?iè
.
3
(5)? úM(x;y) = 2xylny,N(x;y) = x2 +y2p1+y2,??
E = @M@y? @N@x = 2xlny;
[
?
^ ?Z?.?EM =?1yDxí1,yN?Z?μoG ??y￥sy0
(y) = 1y.yNZ?
2xlnydx+(x
2
y +y
p
1+y2)dy = 0
1 ?Z?, |x0 = 0,y0 = 1, V9

U(x;y) =
Z x
0
2xlnydx+
Z y
1
y
p
1+y2dy
= x2 lny + 13(1+y2)32? 23p2:
#?Z?￥Y31x2 lny + 13(1+y2)32 = C, ?C1 ?iè
.
5.
k pBernoulliZ?￥sy0.
3:üBernoulliZ????
T
(a(x)y +f(x)yfi)dx?dy = 0;
? ??￥??f
z = y1?fi,¤
[(1?fi)a(x)z +(1?fi)f(x)]dx?dz = 0:
?
^B?1?z￥B¨L?Z?,? è1.2?
μsy0
0(x) = e?(1?fi)
R a(x)dx
:
'Z?
0(x)[(1?fi)a(x)z +(1?fi)f(x)]dx0(x)dz = 0
1 ?Z?,??dz = (1?fi)y?fidy,??N?Z?
0(x)y?fi[a(x)y +f(x)yfi]dx0(x)y?fidy = 0
1 ?Z?,?"á
ì p
BernoulliZ?￥B?sy0
= y?fie(1?fi)
R P(x)dx
:
8.X?±sZ?
(x2 +y)dx+f(x)dy = 0
μsy0? = x,
k p
μ V
￥f
f(x).
3:
7M(x;y) = x2 +y,N(x;y) = f(x),?
óZ?μsy0? = x?
@(xM)
@y =
@(xN)
@x ;
'x = xf0(x)+f(x),yNf
f(x)
@B¨L?Z?
f0(x) =?f(x)x +1;
p Y3'¤
P
óZ?μsy0? = x￥f
f(x)1
f(x) = Cx + x2;
?C1 ?iè
.