•

•

•

•

•

•

•

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