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 .
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