y = 3x cos2 (y )
,
22
y
sec2 y dy =
tan y =
3x2
+C ,
2
y = tan1
y = 8x3 ey ,
dy
= 3xdx
cos2 y
3xdx
3x2
+C
2
y.
y (1) = 0
ey dy =
8x3 dx
ey = 2x4 + C
y = ln 2x4 + C .
1 = 2 + C
C = 1
y = ln 2x4 1 .
y = y 2 ex
x
y (2) = 1
2
1
=
y
dy
=
y2
t2
e dt
2
2
ex dx
x
x2
2
et dt + C
e dx =
2
y (2) = 1 1 = 0 + C C = 1.
1
=1
y
x
2
et dt,
2
y=
1
1
.
x t2
e dt
2
y=
x2 + 4y
2
x +
dy
y (1 + y )
=
dx
x
y (0) = 0
y (0) = 0
y (0) = 1
y (0) = 1
C
C = 0
x=0
y (1) = 0
y (1) = 0
y=0
C
y=0
x
dy
y
= ry ln
dt
K
= ry
ln
dy
=
y ln (y/K )
K
y
y
K
y
+1 .
K
y = Ke1
y = 0 ln (y/K ) = 1
dy
dy
y
= ry ln
=
y
dt
K
y ln K
y
d2 y
ry
= ry ln
2
dt
K
t
r dt
r dt
u = ln (y/K ) ;
1
du = dy
y
du
= r dt
u
ln |u| = rt + C1
ln |ln (y/K )| = rt + C1
|ln (y/K )| = ert+C1 = eC 1 ert = C2 ert
ln (y/K ) = C2 ert
y = KeC2 e
y<K
C2 C2 = ln (P0 /K )
y = Keln(P0 /K )e
=K
K = 10,
P0 = 0.5,
r = 0.01
C2 = eC1 > 0
rt
y (0) = P0
P0
K
P0
= eC2 ln (P0 /K ) =
K
rt
ert
.