Ecuaciones

8.- (x2 + y2) dx + (x2-xy)dy =0
Y=vx ; x =vy ; dy=vdx+xdv
(x2+v2x2) dx+(x2-vx2)(vdx+xdv)=0X2dx+v2x2 dx+x2vdx+x3dv-v2x2dx-vx3dv=0
Dx+vdx+xdv-xvdv=0
((1+v)dx+(1-v)xdv=0) * 1/(1+v)x
Dx/x+1 - v/1+v dv =0S dx/x- Sv-1/v+1 dv = S0
Ln x – S(1+ -2/v+1)dv=0
Ln x – Sdv + 2 Sdv/v+1 =0
Ln x –v +2 ln l v+1 l =cLn x (v+1)2 =c+v2
Eln[x(/y/x+1)2`] = e c+y/x
X(y/x+1)2= ec * ey/x
X(y/x+1)2 = cey/x
X(y2/x2 +2y/x +1) =cey/x
Y2 /x + 2y + x= ce y/x
Y2 + 2xy + x2 = cxey/x
RESULTADO : (x+y)2 = cxey/x

12.- y2 dx + (x2+xy+y2) dy =0
X=vy ; dx = vdy + ydv
Y2 (vdy + ydv) + (v2y2 + vy2+ y2) dy = 0
Vy2 dy + y3dv + v2y2dy +vy2dy + y2 dy =0
2vy2 dy/y2 + v2y2/y2 dy + y2dy/y2 + y3/y2 dy =0
2v dy + v2 dy + dy + y dy = 0
[(2v + v2+ 1) dy + y dv =0] * 1/(v2 +2v+ 1) (y)
Dy/y +dv / v2 + 2v + 1
Dy / y + dv/ (v+1)2
S dy/y + S (v+1)-2 dv= S0
Ln y + (v+1)-1 / -1 = C
Ln y – 1/ x/y + 1 = C
Ln y – y/ x+y = C
E ln y = e c + y/ x+y
Y= ce y/x+y
1 = ce 1/ 0+1
1 = ce1
1/e = C
Y = 1/ e * e y/ x+y
Y = ey/ x+y / e
RESULTADO : y= ey/ x+y -1