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y+2x+y+1 )
2
dy
dx
= 2( y+2x+y+1 )
2
dz
dw
= 2( z�2+2w+1+z�2+1 )
2
dz
dw
= 2( z
w+z
)2
dz
dw
= u+ w
du
dw
u+ w
du
dw
= 2⇣
uw
w+uw
⌘2
u+ w
du
dw
= 2⇣
u
1+u
⌘2
w
du
dw
= 2u2
1+2u+u
2 � u
w
du
dw
= 2u2
1+2u+u
2 � u
w
du
dw
= 2u2�u�2u2�u
3
1+2u+u
2
1+2u+u
2
�u�u
3du
dw
= dw
w
1+2u+u
2
�u(1+u
2)du
dw
= dw
w
1 + 2u+ u
2 = �A
u
+ Bu+c
(1+u
2)
1 + 2u+ u
2 = A+Au
2 �Bu
2 � Cu
1 + 2u+ u
2 = A+ u
2(A�B)� Cu
�´
du
u
� 2´
du
(1+u
2) =´
dw
w
�ln(u)� 2arctg( 1u
) = ln(w) + c
�ln(uw)� 2arctg( 1u
) = c
�ln( z
w
w)� 2arctg(wz
) = c
�ln(y + 2)� 2arctg(x�1y+2 ) = c
dy
dx
= y�x+1y�x+2
dz
dx
= 1� dy
dx
dy
dx
= 1� dz
dx
dz
dx
= z+1z+2
dz
dx
= 1� z+1z+2
dz
dx
= z+2�z�1z+2
dz
dx
= 2z+2
(z + 2)dz = 2dx
z
2
2 + 2z = 2x
(x�y)2
2 + 2(x� y) = 2x+ c
x
2 � 2xy + y
2 + 4x� 4y � 4x = c
c = x
2 � 2xy + y
2 � 4y
(y+2)(2x+y�4) =
dy
dx
dy
dx
= (z�2+2)2w+6+z�2�4
dy
dx
= z
2w+z
dz
dw
= u+ w
du
dw
u+ w
du
dw
= uw
2w+uw
u+ w
du
dw
= u
2+u
w
du
dw
= u�2u�u
2
2+u
w
du
dw
= �u�u
2
2+u
(2+u)du(�u�u
2) =dw
w
(2+u)du�u(1+u) =
dw
w
2 + u = A
�u
+ B
(1+u)
2 + u = A+ uA� uB
2 + u = u(A�B) +A
1 = A�B
A = 2
B = 1
´2�u
du+´
du
(1+u) =´
dw
w
�2ln(u) + ln(1 + u) = ln(w) + c
ln( 1+u
u
2 ) = ln(w) + c
ln
⇢
1+ z
w
z
2
w
2
�
= ln(x� 3) + c
ln
✓
w+z
z
2w
◆
= ln(x� 3) + c
ln
⇣
(x�3+y+2)(x�3)(y+2)2
⌘
� ln(x� 3) = ln(c)
ln(x+y�1(y+2)2 ) = ln(c)
c = (x+y�1)(y+2)2
y
0 = x�2y+5y�2x�4
dy
dx
= x�2y+5y�2x�4
1)x� 2y = �5
2)� 2x+ y = 4
dz
dw
= w�1�2z�4+5z+2�2w+2�4
dz
dw
= w�2zz�2w
dz
dw
= u+ w
du
dw
u+ w
du
w
= w�2uwuw�2w
u+ w
du
w
= 1�2uu�2
w
du
dw
= 1�2uu�2 � u
w
du
dw
= 1�2u�u
2+2uu�2
w
du
dw
= 1�u
2
u�2
(u�2)(1�u
2)du = dw
w
(u�2)(1�u)(1+u)du = dw
w
u� 2 = A
(1�u) +B
(1+u)
u� 2 = A(1 + u) +B(1� u)
u� 2 = A+ uA+B � uB
u� 2 = A+B + u(A�B)
�2 = A+B
1 = A�B
� 12 = A
32 = B
12
´du
(1�u) �32
´du
(1+u) =´
dw
w
12 ln(1� u)� 3
2 ln(1 + u) = ln(w) + ln(c)
� 12 {ln(1� u) + 3ln(1 + u)} = ln(w) + ln(c)
12 ln
�
(1� u)(1 + u)3
= ln(w) + ln(c)
12 ln
�
(w�z
w
)(w+z
w
)3
= ln(x+ 1) + ln(c)
12 ln
n
(x+1�y+2x+1 )(x+1+y�2
x+1 )3o
= ln(x+ 1) + ln(c)
12
n
(x�y+3)(x+y�1)3
(x+1)3
o
x
3(y0 � x) = y
2
y = z
↵
y
0 = ↵z
↵�1z
0
x
3(y0 � x) = y
2
y
0 = ↵z
↵�1z
0
x
3�
↵z
↵�1z
0 � x
�
� y
2 = 0
x
3↵z
↵�1z
0 � x
4 � z
2↵ = 0
3 + ↵� 1 = ↵+ 2
2↵
↵+ 2 = 4
↵ = 2
↵
2x3zz
0 � x
4 � z
4 = 0
z
0 x
4+z
4
2x3z
dz
dx
= u+ x
du
dx
u+ x
du
dx
= x
4+u
4x
4
2ux4
x
du
dx
= 1+u
4
2u � u
x
du
dx
= 1+u
4�2u2
2u
2u1+u
4�2u2 du = dx
x
2u(u2�1)2 du = dx
x
m = u
2 � 1
dm = 2udu´
dm
m
2
´dx
x
� 1m
= ln(x) + c
� 1u
2�1 = ln(x) + c
x
2
z
2�x
2 = ln(x) + c
y = z
2
z =py
x
2
y�x
2 � ln(x) = c
2y0 + x = 4py
y = z
↵
z
0 = ↵z
↵�1z
0
2(↵z↵�1z
0) + x = 4pz
↵
↵� 1
1
↵
2
↵� 1 = 1
↵ = 2
4zz0 + x = 4z
z
0 = 4z�x
4z
z
0 = u+ x
du
dx
u+ x
du
dx
= 4ux�x
4ux
u+ x
du
dx
= 4u�14u
x
du
dx
= 4u�14u � u
x
du
dx
= 4u�1�4u2
4u
4u�4u2+4u�1 = dx
x
4u = � 12 {(�8u+ 4)� 4)}
12
´ �8u+4�4u2+4u�1du+ 2
´du
�4u2+4u�1 =´
dx
x
12
´ �8u+4�4u2+4u�1du� 1
2
´du
(u� 12 )
2+�p
32
�2 =´
dx
x
12 ln(�4u2 + 4u� 1)� 1
2arct
✓
u� 12p
32
◆
= ln(x) + c
12 ln(�4 z
2
x
2 + 4 z
x
� 1)� 12arct
✓
2z�z
2xp3
2
◆
= ln(x) + c
12 ln(�4 y
x
2 + 4py
x
� 1)� 12arct
⇣
2py�xp3x
⌘
� ln(x) = c
23xyy =
p
x
6 � y
4 + y
2
y = z
↵
y
0 = ↵z
↵�1z
023xz
↵
↵z
↵�1z
0 =px
6 � z
4↵ + z
2↵
1 + 2↵� 16� 4↵2↵
! 6↵ = 6↵ = 1
) 23xzz
0 = z
2 +px
6 � z
4
z = ux
z
0 = u+ x
du
dx
y( 1px
) =´( 1p
x
)(x2�12 )dx+ c´
( 1px
)(x2�12 )dx+ c = 1
2
´x
3/2dx� 1
2
´dxpx´
( 1px
)(x2�12 )dx+ c = x
5/2
5 � ln(px) + c
y( 1px
) = x
5/2
5 � ln(px) + c
y =px
3
5 � ln(px)p
x
+ cpx
(x2y
2 + 1)ydx+ (x2y
2 � 1)xdy = 0
y = z
↵
y
0 = ↵z
↵�1z
0
dy
dx
= � (x2y
2+1)y(x2
y
2�1)x
↵z
↵�1z
0 = � (x2z
2↵+1)z↵
(x2z
2↵�1)x
(↵z↵�1z
0)((x2z
2↵ � 1)x) = �(x2z
2↵ + 1)z↵
↵z
3↵�1x
3z
0 � x↵z
↵�1z
0 + x
2z
3↵ + z
↵ = 0
3↵� 1 + 3
1 + ↵� 1
2 + 3↵
↵
3↵+ 2 = ↵
↵ = �1
! �z
�4x
3z
0 + xz
�2z
0 + x
2z
�3 + z
�1 = 0dz
dx
(�z
�4x
3 + xz
�2) = �x
2z
�3 � z
�1
z
0 = u+ x
du
dx´1
(lnx)2 (1x
)dx+ c
m = lnx; dm = dx
x
´dm
m
2 = � 1m
+ c
´dm
m
2 = � 1ln(x) + c
y( 1(lnx)2 ) =
´1
(lnx)2 (1x
)dx+ c
y( 1(lnx)2 ) = c� 1
ln(x)
(2x+ 4y) + (2y � 2)y‘ = 0
(2x+ 4y)dx� (2x� 2y)dy = 0
M(x, y) = (2x+ 4y)
My = 4
N(x, y) = (2y � 2)
Nx = 2
My 6= Nx
2x(1 +p
x
2 � y)dx�p
(x2 � y)dy = 0
(2x+ 2xp
x
2 � y)dx�p
(x2 � y)dy = 0
M(x, y) = (2x+ 2xp
x
2 � y)
My = �x
p
(x2 � y)
N(x, y) = �p
(x2 � y)
Nx = �x
p
(x2 � y)
My = Nx
@f(x,y)@,x
= (2x+ 2xp
x
2 � y)
f(x, y) =´2x+ 2x
p
x
2 � y + C
f(x, y) = x
2 + 23 (x
2 � y)32 + h(y)
@f(x,y)@,y
= x
2 + 23 (x
2 � y)32 + h(y)
@f(x,y)@,y
= �p
x
2 � y + h0(y)
�p
x
2 � y + h0(y) = �p
x
2 � y
h0(y) = 0
h(y) = C
C = x
2 + 23 (x
2 � y)32
(exSen(y)� 2tSen(x))dx+ (exCos(y) + 2Cos(x))dy = 0
(exSen(y)� 2tSen(x))dx+ (exCos(y) + 2Cos(x))dy = 0
M(x, y) = (exSen(y)� 2Sen(x)
My = e
x
Cos(y)� 2Sen(x)
N(x, y) = e
x
Cos(y) + 2Cos(x)
Nx = e
x
Cos(y)� 2Sen(x)
My = Nx
@f(x,y)@,x
= e
x
Sen(y)� 2ySen(x)
f(x, y) =´e
x
Sen(y)� 2ySen(x)dx+ C
f(x, y) = e
x
Sen(y) + 2yCos(x)) + h(y)
@f(x,y)@,y
= e
x
Sen(y) + 2yCos(x)) + h(y)
@f(x,y)@,y
= e
x
Cos(y) + 2Cos(x)) + h0(y)
e
x
Cos(y) + 2Cos(x)) + h0(y) = e
x
Cos(y) + 2Cos(x)
h0(y) = 0
h(y) = C
C = e
x
Sen(y) + 2yCos(x))
(xln(y) + xy)dx+ (yln(x) + xy)dy = 0
(xln(y) + xy)dx+ (yln(x) + xy)dy = 0
M(x, y) = xln(y) + xy
My = x
y
+ x
N(x, y) = yln(x) + xy
Nx = y
x
+ x
My 6= Nx
(x3 � 3xy2)dx+ (y3 � 3x2y)dy = 0
(x3 � 3xy2)dx+ (y3 � 3x2y)dy = 0
M(x, y) = x
3 � 3xy2
My = �6xy
N(x, y) = y
3 � 3x2y
Nx = �6xy
My = Nx
@f(x,y)@,x
= x
3 � 3xy2
f(x, y) =´x
3 � 3xy2dx+ C
f(x, y) = x
4
4 � 32x
2 + h(y)
@f(x,y)@,y
= x
4
4 � 32x
2 + h(y)
@f(x,y)@,y
= �3x2y + h0(y)
�3x2y + h0(y) = y
3 � 3x2y
h0(y) = y
3
h(y) = y
4
4
x
4
4 � 32x
2 + y
4
4 = C
h
1 + exp(xy
)i
dx+ exp(xy
)(1� x
y
)dy = 0
My = �xe
x
y
y
2
Mx = � e
x
y
y
+ (1� x
y
)ex
y
1y
= �xe
x
y
y
2
Mx = My,
@f(x,y)@x
= 1 + e
x
y
f(x, y) =´1 + e
x
y
dx+ c
f(x, y) = x+ ye
x
y + c; c = h(y)
@f(x,y)@y
= �xye
x
y
y
2 + e
x
y + h(y) = e
x
y (1� x
y
)
h(y) = e
x
y � e
x
y
x
y
� e
x
y + e
x
y
x
y
h(y) = 0 h(y) = c1
! f(x, y) = x+ ye
x
y + c
y + ay = e
mx
dy
dx
+ ay = e
mx
F.I = u = e
´adx
u = e
ax
! ye
ax =´(eaxemx)dx+ c
ye
ax = e
ax+mx
a+m
+ c m 6= �a
(2x+ 1)y=4x+ 2ydy
dx
(2x+ 1)� 2y = 4x
dy
dx
� 22x+1y = 4x
2x+1
F.I = u = e
�´
22x+1dx = 1
2x+1
! y
12x+1 =
´( 12x+1 )(
4x2x+1 )dx+ c
y
12x+1 = 1
2x+1 + ln(2x+ 1) + c
y = 1 + (2x+ 1) [ln(2x+ 1) + c]
x
2y
0 + xy + 1 = 0
x
2 dy
dx
+ xy + 1 = 0dy
dx
+ 1x
y = � 1x
2
F.I = u = e
´1x
dx = x
! yx =´x(� 1
x
2 )dx+ c
xy = �ln(x) + c
y = 1x
[c� ln(x)]
dy
dx
� 1) = 2yln(x)
x
dy
dx
= 2yln(x) + 1
dy
dx
= 2yxln(x) +
1x
dy
dx
� 2yxln(x) =
1x
u = e
´P (x)dx
u = e
´ �2xlnx
dx
m = lnx; dm = dx
x
u = e
´ �2dmm
dx
u = e
�2ln(lnx)
u = e
ln(lnx)2
u = 1(lnx)2
y ⇤ u =´u ⇤Q(x)dx+ c
y( 1(lnx)2 ) =
´1
(lnx)2 (1x
)dx+ c
´1
(lnx)2 (1x
)dx+ c
m = lnx; dm = dx
x´dm
m
2 = � 1m
+ c
´dm
m
2 = � 1ln(x) + c
y( 1(lnx)2 ) =
´1
(lnx)2 (1x
)dx+ c
y( 1(lnx)2 ) = c� 1
ln(x)
y = cln
2x� ln(x)
2
dy
dx
= 1sen
2y+xctg(x)
dx
dy
= sen
2y + xctg(x)
dx
dy
� xctg(y) = sen
2y
dx
dy
� x
cos(y)sen(y) = sen
2y
u = e
´P (y)dy
u = e
�´
cos(y)sen(y)dy
sen(y) = m; cos(y) = dm
u = e
�´
dm
m
u = e
�ln(m)
u = e
ln(sen(y))�1
u = 1sen(y)
x ⇤ u =´u ⇤Q(y)dy + c
x( 1sen(y) ) =
´1
sen(y) (sen2y)dy + c
´1
sen(y) (sen2y)dy + c
´sen(y)dy + c = �cos(y) + c = c� cos(y)
x( 1sen(y) ) =
´1
sen(y) (sen2y)dy + c
x( 1sen(y) ) = c� cos(y)
x= sen(y) [c� cos(y)]
dy
dx
= y(y�1)(1�2xy)
dx
dy
= 1y
2�y
� 2x(y�1)
dx
dy
+ 2x(y�1) =
1(y2�y)
u = e
´P (y)dy
u = e
´2
y�1dy
u = e
2ln(y�1)
u = (y � 1)2
x ⇤ u =´u ⇤Q(y)dy + c
x ⇤ (y � 1)2 =´(y � 1)2 ⇤ 1
y(y�1)dy + c
´(y � 1) ⇤ 1
y
dy + c
´dy �
´dy
y
= y � ln(y) + c
x ⇤ (y � 1)2 =´(y � 1)2 ⇤ 1
y(y�1)dy + c
x ⇤ (y � 1)2 = y � ln(y) + c
x = 1(y�1)2 (y � ln(y) + c)
4
u = e
´P (x)dx
u = e
�´
2x
dx
u = e
�2ln(x)dx
u = 1x
2
y ⇤ u =´u ⇤Q(x)dx+ c
y( 1x
2 ) =´
1x
2 (x4)dx+ c
´(x2)dx+ c = x
3
3 + c
y( 1x
2 ) =´
1x
2 (x4)dx+ c
y = (x2)h
x
3
3 + c
i
3
2x dy
dx
� y = x(x2 � 1)
2x dy
dx
= x
3 � x+ y
dy
dx
= (x2 � 1) 12 + y
12x
dy
dx
� y
2x = x
2�12
u = e
´P (x)dx
u = e
´� dx
2x
u = e
ln(x)1/2
u = 1px
y ⇤ u =´u ⇤Q(x)dx+ c
y( 1px
) =´( 1p
x
)(x2�12 )dx+ c
´( 1p
x
)(x2�12 )dx+ c = 1
2
´x
3/2dx� 1
2
´dxpx
´( 1p
x
)(x2�12 )dx+ c = x
5/2
5 � ln(px) + c
y( 1px
) = x
5/2
5 � ln(px) + c
y =px
3
5 � ln(px)p
x
+ cpx
(x2 � y
2 � 1)y0 = 2xy
dx
dy
=x
2 � y
2 � 1
2xy
dx
dy
� 1
2y(x) =
�y
2 � 1
2xy
P (y) =1
2y�! µ(y) = e
´(�1
2y )dy
u(y) =1py
x
py
=
ˆ1py
✓
�y
2x� 1
2xy
◆
dy + C
x
py
= �y
32
3x� 1
pyx
+ C
x
2 =�y
32y
12
3�
py
pyx
+ C
y
2 = 3(Cx� x
2 � 1
x
)
y =
r
3(Cx� x
2 � 1
x
y
0 � y = 2x� x
2
P (x) = �1 �! u(x) = e
´�dx
u(x) = �x
ye
�x =
ˆe
�x(2x� x
2)dx
ye
�x =
ˆe
�x2xdx�ˆ
e
�x
x
2dx
ye
�x = �2e�x(x+ 1) + e
�x(x2 + 2x+ 2) + C
y = �2x� 2 + x
2 + 2x+ 2 + Ce
x
y = x
2 + Ce
x
y
0 + 2y = x
3
P (x) = 2
u = e
´2dx �! u(x) = e
2x
ye
2x =
ˆe
2xx
3dx+ C Integrando por partes
ye
2x =x
3e
2x
2� 3
2
ˆe
2xx
2dx+ C Integrando por partes
ye
2x =x
3e
2x
2� 3
4(x2
e
2x) + 3
ˆe
2xxdx+ C Integrando por partes
ye
2x =x
3e
2x
2� 3
4(x2
e
2x) +3
4xe
2x � 3
8e
2x + C
y =x
3
2� 3x2
4+
3
4x� 3
8+ e
2xC
y
0 + y = sin(x) + cos(x)
P (x) = 1
u(x) = e
´dx �! u(x) = e
x
ye
x =
ˆe
x(sinx+ cosx)dx+ C
ˆe
x
sinx = e
x
sinx�ˆ
e
x
cosxdx
ˆe
x
sinx = e
x
sinx� e
x
cosx�ˆ
e
x
sinxdx
ˆe
x
sinx =1
2e
x(sinx� cosx)
ˆe
x
cosx =1
2e
x(sinx+ cosx)
ye
x = e
x
sinx+ C
y = sinx+C
e
x
y
0 � 2xy = 1
P (x) = �2x
u(x) = e
´�2xdx �! u(x) = e
�x
2
ye
�x
2
=
ˆe
�x
2
dx+ C
ye
�x
2
=
ˆe
u
dupu
+ C
y = e
x
2
(
ˆe
�x
2
dx+ C)
(x� 1)( dydx
+ y
2) = �y
dy
dx
+ y
x+1 = �y
2
p = 1x+1 ;n = 2
z
0+ z
x+1 (1� 2) = �1(1� 2)
z
0+� z
x+1 = 1´e
´� 1
x+1dx+C
e
�´� 1
x+1dx´e
�ln(x+1)+C
e
�ln(x+1)
z = 1x+1 + C(x+ 1)
z = [ln(x+ 1) + C](x+ 1)
z
0 � 4x
(1� 1/2) = 2x(1� 1/2)
y
2 = (x+ 1)ln(x+ 1) + C
xy
0 � 2x2py = 4y
z = 1x+1 + C(x+ 1)
y
0 � 2xpy = 4y
x
p = � 4yx
, q = 2x;n = 1/2
y
0 � 2xpy = 4y
x
p = � 4x
, q = 2x, n = 1/2
z
0 � �2x
= x
z = [´xe
� 2x
dx+C ]e�´� x
y
dx
z = [´xe
�2ln(x)+C ]e�2ln(x)dx
z = [´xe
�2ln(x)+C ]e�2ln(x)dx
z = [x ⇤ 1x
2 + C]x2
z = [ln(x) + C]x2
z = x
2ln(x) + x
2C
py = x
2ln(x) + C
2y0 � x
y
= xy
x
2�1
y
0 � x
2(x2�1)y = xy
�1
p = � x
2(x2�1) , q = x;n = �1
z
0 � x
2(x2�1) (1 + 1) = x(1 + 1)
z
0 � x
(x2�1) = 2x
z = [´2xe
x
x
2�1dx+C
]e�´
x
x
2�1dx
z = [´2xe1/2ln(x
2�1) + C]e�1/2ln(x2�1)
z = [2xpx
2 � 1 + C] 1px
2�1
z = [2(x2 � 1)3/2 + C
C
(x2�1)3/2]
1y
= [2(x2 � 1)3/2 + C
C
(x2�1)3/2]
y = [ 3(px
2�1)2(x2�1)+3C ]
(2x2ylny � x)y
0= y
y
0= y
2x2ylny�x
y
0= y
2xylny�1x
y
0= 1
xlny�1 � y
x
y
0+ y
x
= 1xlny�1 ;n = 0
z
0+ 1
x
(1� 0) = 1xlny�1 (1� 0)
z
0+ 1
x
(1) = 1xlny�1 (1)
z = [´
1xlny�1e
´1x
dx
dx+ C]e�´
1x
dx
z = [ x
lny�1 + C] 1x
z = 1lny�1 + k
x
y
0 = 1lny�1 + k
x
1 = 1lny�1 + k
x
y
0 � y + y
2(x2 + x+ 1) = 0
y
0 � y = �(x2 + x+ 1)y2
p = �1, q = �(x2 + x+ 1);n = 2
z
0 � (1� 2)(�1)z = �(x2 + x+ 1)(1� 2)
z
0 � z = x
2 + x+ 1
p = �1, q = (x2 + x+ 1)
z = [´(x2 + x+ 1)e�
´dx + C]e�
´�dx
z = [´(x2 + x+ 1)e�x + C]e�x
z = (x2 + x+ 1)e�x � (2x+ 1)e�x + 2´e
�x
dx
z = (x2 + x+ 1)e�x � (2x+ 1)e�x + 2e�x
z = [e�x(x2 + x+ 1� 2x� 1� 2) + C]e�x; e0 = 1
z = (x2 � x� 2) + Ce
x
y
2 = (x2 � x� 2) + Ce
x
y
0 + 2y = 2py
y
0 + 2y = 2(y12 )
w = y
12
dw
dx
+�
12
�
(2)w =�
12
�
(2)
dw
dx
+ w = 1
f
i
= e
´dx = e
x
we
x =´e
x
dx+ c
we
x = e
x + c
w = 1 + ce
�x
y
12 = 1 + c
e
x
= e
x+C
e
x
y = (ex+c)2
e
2x
y
0 + 2xy = 2x3y
3
w = y
�2
dw
dx
+ (�2)(2x)w = (�2)(2x3)
dw
dx
� 4xw = �4x3
f
i
= e
´�4xdx = e
�2x2
we
�2x2
=´e
�2x2
(�4x3)dx+ c
we
�2x2
= �2´e
�2x2
(2x3)dx+ c
�2´e
�2x2
(2x3)dx�
u = x
2
= �2´e
�2uudu
�
f = u dg = e
�2udu
2e2u
= e
�2uu�´e
�2udu
{s = �2u
e
�2uu+ 1
2
´e
s
ds = e
s
2 + e
�2uu
= e
�2u
2 + e
�2uu = e
�2x2
x
2 + 12e
�2x2
we
�2x2
= e
�2x2
x
2 + 12e
�2x2
+ c
w = x
2 + 12 + ce
2x2
= y
�2
y = ±q
22x2+ce
2x2+1
y
n�1(ay0 + y) = x
y
0 + y
a
= x
a
y
1�n
w = y
n
dw
dx
+ n
a
w = n
a
x
f
i
= e
´n
a
dx = e
n
a
we
nx
a =´e
nx
a
nx
a
dx+ c
n
a
´xe
nx
a
dx
�
f = x dg = e
nx
a
dx
ne
nx
a
= xe
nx
a �´e
nx
a
dx
�
u = nx
a
adx
= xe
nx
a � a
n
´e
u
du = xe
nx
a � a
n
e
u
= xe
nx
a � a
n
e
nx
a
we
nx
a = xe
nx
a � a
n
e
nx
a + c
w = x� a
n
+ ce
�nx
a = y
n
y =⇣
x� a
n
+ ce
�nx
a
⌘
1n
�
xy + x
2y
3�
y
0 = 1
y
0 = 1(xy+x
2y
3)
dx
dy
= xy + x
2y
3
dx
dy
� xy = x
2y
3
w = x
�1
dw
dy
+ (�1)(�y)w = (�1)y3
dw
dy
+ yw = �y
3
f
i
= e
´ydy = e
y
2
2
we
y
2
2 =´e
y
2
2 (�y
3)dy + c
´e
y
2
2 (�y
3)dy
�
u = y
2
= � 12
´e
u
2udu
�
f = u dg = e
u
2du
u
2
= �e
u
2u+ 2
´e
u
2du
�
s = u
2
2du
= �e
u
2u+ 2
´e
s
ds = 2es � e
u
2u
= 2eu
2 � e
u
2u = 2e
y
2
2 � e
y
2
2y
2 + c
we
y
2
2 = 2ey
2
2 � e
y
2
2y
2 + c
w = 2� y
2 + ce
�y
2
2 = x
�1
x = 1
2�y
2+ce
�y
2
2
y
0 = y
x+y
2
dx
dy
= x
y
+ y
dx
dy
� x
y
= y
u = e
´� 1
y
dy = e
�ln(y) = 1y
x
y
=´dy + c
x
y
= y + c
x = y
2 + cy