1
52.1
1.Z?(2.1.1)? ?T
àμL
!g(y) 6= 0,)
8"¨s ?M
E ? p3±sZ?.
3:á
ìs/
? f? ?)
Z?(2.1.1)￥3. ?Tg(y0) = 0,5y = y0A ?
^Z?(2.1.1)
￥3. ?Tg(y0) 6= 0.
!y = ’(x) uW(a;b)

^
@
SHq’(x0) = y0￥Z?(2.1.1)￥3
5
d’(x)
dx = h(x)g(’(x)); 8a < x < b:
?3￥·B? V? uW(a;b)
 (μg(’(x)) 6= 0.
Y
L
L
!μ?x0 2 (a;b)
P¤g(’(?x0)) =
0,5y =?(x) · ’(?x0) (èf
)
^Z?(2.1.1)￥3.V7f
’,??
^V?(?x0;’(?x0))￥Z?
(2.1.1)￥3.?3￥·B?,’ ·? · ’(?x0).#
g(y0) = g(’(x0)) = g(’(?x0)) = 0:
?DL
!g(y0) 6= 0
±.
?g(’(x)) 6= 0 V¤
1
g(’(x)) ￠
d’(x)
dx = h(x):
#?x 2 (a;b)
H
Z x
x0
’0(t)
g(’(t))dt =
Z x
x0
h(t)dt:
7? = ’(t),5
Z ’(x)
’(x0)
d?
g(?)d? =
Z x
x0
h(t)dt;
#y = ’(x)
^
@
Z y
y0
d?
g(?) =
Z x
x0
h(t)dt; ’(x0) = y0;
￥?f
3.
Q-, ?y =?(x)
^?

T
??￥?f
5y =?(x)
^V(x0;y0)￥Z?(2.1.1)￥3.
Y
L
,?

T??a < x < b
H,
Z?(x)
y0
d?
g(?) =
Z x
x0
h(t)dt:
H1?x p?¤?
1
g(?(x)) ￠
d?(x)
dx = h(x):
'
d?(x)
dx = h(x)g(?(x)):
[y =?(x)
^Z?(2.1.1)￥3.
2
] ? V￡??
T
Z y
y0
d?
g(?) =
Z x
x0
h(t)dt+C;
??￥f
y =?(x;C)?
^Z?(2.1.1)￥3, ?C
^ ?iè
.
yN,
L= p3?"
 p
Pg(y) = 0￥y′[?,o1¨g(y)"Z?(2.1.1)￥
H ?a
p??s
Z dy
g(y) =
Z
h(x)dx;
' V.
2.
k¨s ?M
E p/
B¨±sZ?￥3.
(1) dydx =?xy:
(3) dydx = 2xy:
(4) xy(1+x2)dy = (1+y2)dx:
(9) dydx =
p
1?y2p
1?x2
:
(12) dydx = cosx3y2+ey:
3:
(1)s ?M
a¤ydy =?xdx,
Hs,¤
y2
2 =?
x2
2 +C1;
y7eZ?￥Y31
x2 +y2 = C;
?C = 2C11 ?idμè
.
(3)?y 6= 0
H,s ?M
a¤
1
ydy = 2xdx;

s¤
lnjyj = x2 +C1;
N?A ?y = 09
^Z?￥3.V7Z?￥Y31
y = Cex2;
?C1 ?iè
.
(4)s ?M
a¤
ydy
1+y2 =
dx
x(1+x2);

s¤1
2 ln(1+y
2) = lnjxj? 1
2 ln(1+x
2)+C1;
'
ln(1+x2)(1+y2) = lnx2 +2C1;
V7Z?￥Y31
(1+x2)(1+y2) = Cx2;
?C = e2C11 ?i?è
.
3
(9)?y 6= §1
H,s ?M
a¤
dyp
1?y2 =
dxp
1?x2;

s¤
arcsiny = arcsinx+C;
?C1 ?i?è
,V7Z?￥Y31
y = sin(arcsinx+C);
?C1 ?iè
.N?A ?y = 1y =?19
^Z?￥3.
(12)s ?M
a¤
(3y2 +ey)dy = cosxdx;

s¤Z?￥?
TY3
y3 +ey = sinx+C;
?C1 ?iè
.