65.( x2eyx+xy )dy

dx=xy e

yx+x2+ y2 , y (e )=e

64.dydx

= yx+ xy

x2−xy+ y2, y (2 )=0

ttt ∙ t

t 2t ∙ t t 2

1t 2

t 2t 2t 2

(x2 eyx +xy )dy=( xy e

yx +x2+ y2)dx

t 2t ∙ t t ∙ t t 2t 2

t 2t 2t 2t 2t 2homogénea

dy=vdx+xdv y=vx v= yx

( x2 evxx +x ( vx )) vdx+xdv=(x (vx )e vxx +x2+v2 x2)dx

(x2 ev+x2 v )vdx+xdv=(x2 vev +x2+v2 x2 )dx

x2 ev vdx+x3 evdv+x2 v2dx+ x3 vdv=x2 vev dx+x2dx+v2 x2dx

[ x3 evdv+x3 vdv=x2dx ] 1x3

∫ evdv+∫ vdv=∫ 1x dx

[ev+ v22 =lnx+c ]22ev+v2=ln x2+c

2ev+v2−ln x2=c ----sol gral.

2eyx+ yx22

=ln x2+c

2eee+ ee22

−ln e2=c

No homogénea

2e+1−2=2e−1

2e−1=c

2eyx+ yx22

=ln x2+2e−1

2eee+ ee22

=ln e2+2e−1

2e+1=ln e2+2e−1 ---- Sol. Part.

66.( y2 ln xy +xy )dx=( xyln xy +x2+ y2)dy , y (1 )=1

t 2 ∙ttt ∙ t t ∙ t ∙

ttt 2 t2

t 2t 2t 2t 2t 2−−−−homogénea

dx=vdy+ ydv x=vy v= xy

( y2 ln vyy + (vy ) y) vdy+ ydv=((vy ) yln vyy +v2 y2+ y2)dy( y2 lnv+v y2 )vdy+ ydv=(v y2 lnv+v2 y2+ y2 )dy

y2 vlnv dy+ y3 lnv dv+v2 y2dy+vy3dv= y2 vlnv dy+v2 y2dy+ y2dy

[ y3 lnv dv+vy3dv= y2dy ] 1y3

∫ lnv dv+∫ vdv=∫ 1ydy

v lnv−v+ v2

2=lny+c

xylnxy− xy+

xy2

2

2=lny+c

xylnxy− xy+

xy2

2

2−lny=c−−−−Sol . gral .

11ln11−11+

1122

2−ln 1=c

c=−12

ln 1−1+ 12=ln1−1

2 ----Sol. Part.

67. xdydx

=√ x2+ y2+ y , y (2 )=4

x dy=(√x2+ y2+ y ) dx

t √t 2√ t 2

t t t−−−−homogénea

dy=vdx+xdv y=vx v= yx

x (vdx+xdv )=(√x2+v2 x2+vx )dx

xvdx+x2dv =√ x2+v2 x2dx+vx dxx2dv =√ x2+v2 x2

[x2dv=x √(1+v2 )dx ] 1

x2√(1+v2 )

∫ dv

√1+v2=∫ 1x dx

ln (v+√ (1+v2))−lnx=c

[ ln( v+√(1+v2 )x )=c]

e

v+√ (1+v2 )x

=c

v+√(1+v2 )=cx

yx+√(1+ yx2

2)x

=c−−−−Sol .Gral .

42+√(1+ 422

2)2

=c

c=2+√52

yx+√( x2+ y2x2 )

x=c

y+√ y2+x2=c x2

y+√ y2+x2=( 2+√52 ) x2

4+√20=( 2+√52 )∙ 4−−−−Sol .Part .

68. y2dxdy

=xy+√ x2 y2+x4 , y (0 )=1

t 2t ∙ t √ t 2√ t 2 √ t 4

t 2t 2t ∙ t t 2

t 2t 2t 2t 2−−−−homogénea

y2dx=(xy+√x2 y2+x4 ) dy

dx=vdy+ ydv x=vy v= xy

( y2 )vdy+ ydv=( (vy ) y+√ (vy )2 y2+ (vy )4 ) dy

y2 vdy+ y3dv=( y2 v+√v2 y 4+v4 y 4 )dyy2 vdy+ y3dv= y2 vdy+√v2 y4+v4 y4dyy3dv=√ v2 y4(1+v2)dy

[ y3dv= y2 v √(1+v2)dy ] 1

y3 v √(1+v2)

∫ 1

v √(1+v2)dv=∫ 1

ydy

−arctanh (√ (1+v2 ))=lny+c

−arctanh (√ (1+v2 ))−lny=c−−−−sol . gral

−arctanh (√(1+ x2

y2 ))−lny=c−arctanh (√( x

2+ y2

y2 ))−lny=c−arctanh (√( 0

2+12

12 ))−ln1=c−arctanh (√ (1+v2 ))=−arctanh (√1 )−−−−Sol . Part .

69.dydx

=2 x2 y+ y3

x3, y (1 )=1

t2 ∙t+t 3

t3

t3+t 3

t 3−−−−homogénea

(x3 )dy=(2 x2 y+ y3 )dx

dy=vdx+xdv y=vx v= yx

(x3 )vdx+xdv=(2x2(vx)+(vx )3 )dx

x3 vdx+x4dv=2 x3 v dx+v3 x3dx

[ x4dv=x3dx (v+v3 ) ] 1

x4 (v+v3 )

∫ 1

(v+v3 )dv=∫ 1x dx

[ ln v− ln (v2+1 )2

=ln x+c ]2ln v2−ln (v2+1 )=ln x2+c

ln ( v2v2+1 )=ln x2+c

ln ( v2

v2+1x2

)=ce[ln( v2

v2+1x2

)=c]v2

v2+1x2

=c−−−−sol . gral.

( yx )2

( yx )2

+1

x2=

y2

y2+x2

x2= y2

x2 ( y2+x2 )=c

11 (1+1 )

=12=c

e[ ln( y2

x2( y2+ x2))=c]y2

x2 ( y2+x2 )=12

2 y2=x4+x2 y2

2 (1 )2=14+1212----Sol part.

2=2

70.dydx

=5 x+ yx

t tt−−−−−homogénea

( x )dy=(5 x+vx )dx

dy=vdx+xdv y=vx v= yx

( x ) vdx+xdv=(5 x+vx )dx

xvdx+x2dv=5 xdx+vxdx

[ x2dv=5 xdx ] 1x2

∫ dv=∫ 5 xx2dx

v=5∫ 1xdx

v=5 ln x+c

v−5 ln x=c−−−Sol . gral .

yx−5 ln x=c

1−5 ln1=1

C=1

1=5 ln x 1+1−−−−Sol .Part .

71.yxdxdy

= yxexy+1 , y (0 )=1

dxdy

= yxexy+ xy

tttt−−−homogénea

dx=( yx exy+ xy )dy

dx=vdy+ ydv x=vy v= xy

vdy+ ydv=( yvy evyy + vy

y )dy

vdy+ ydv=( 1v ev+v )dyvdy+ ydv=1

vevdy+vdy

[ ydv= 1v e−v

dy] v e−vy∫ v e−v dv=∫ 1

ydy

−e−v v−e−v=ln y+c

−v+1ev

−ln y=c−−−−Sol . gral .

− xy+1

exy

−ln y=c

−1e0

−ln 1=c

C=-1

−e− xy =ln y−1

−e0=ln1−1−−−−Sol . part .

72. x3dydx

=x2 y+ y3 , y (1 )=1

x3dy=(x2 y+ y3 )dx

t 3t 2 ∙ t t 3

t 3t 3t 3−−−homogénea

dy=vdx+xdv y=vx v= yx

x3 ( vdx+xdv )=(x2(vx)+v3 x3 )dx

x3 vdx+x4dv=x3 vdx+v3 x3dx

[ x4dv=v3 x3dx ] 1

x4 v3

∫ 1

v3dv=∫ 1x dx

−12v2

=ln x+c

−12v2

−ln x=c−−−−sol . gral .

−1

2y2

x2

−ln x=c

−12

−ln 1=−¿ 12¿

C=−12

−12

=ln 1−12−−−−Sol . Part .