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Solve-in-the-order-of-finding-the-integrsting-stroke-1-ydx-x-3-y-x-xdy-0-2-xy-2-y-dx-xdy-0-




Question Number 124763 by Mammadli last updated on 05/Dec/20
Solve in the order of finding the integrsting stroke:  1. ydx−(x^3 y+x)−xdy=0  2. (xy^2 +y)dx−xdy=0
Solveintheorderoffindingtheintegrstingstroke:1.ydx(x3y+x)xdy=02.(xy2+y)dxxdy=0
Commented by Mammadli last updated on 05/Dec/20
Sorry dear ser.. 1. ydx-(x³y+x)dy=0
Answered by benjo_mathlover last updated on 05/Dec/20
2. (dy/dx) = ((xy^2 +y)/x) = y^2 +(y/x)   put v = (y/x)⇒y=vx and (dy/dx) = v + x (dv/dx)  ⇔v + x (dv/dx) = v^2 x^2  + v   ⇒  x (dv/dx) = v^2 x^2  ⇒ (dv/v^2 ) = x dx   ∫ (dv/v^2 ) = ∫ x dx ⇒ −(1/v) = (1/2)x^2 +c  −(1/v) = ((x^2 +C)/2) ; −(x/y)=((x^2 +C)/2)   y = −((2x)/(x^2 +C))
2.dydx=xy2+yx=y2+yxputv=yxy=vxanddydx=v+xdvdxv+xdvdx=v2x2+vxdvdx=v2x2dvv2=xdxdvv2=xdx1v=12x2+c1v=x2+C2;xy=x2+C2y=2xx2+C
Commented by Mammadli last updated on 05/Dec/20
thk you dear ser, please: ydx-(x³y+x)dy=0
Answered by liberty last updated on 06/Dec/20
(1) y dx −(x^3 y+x)dy=0   let x = yv ⇒dx = v dy + y dv  ⇔y(v dy +y dv)−(y^3 v^3 +yv)dy=0  ⇔ y^2  dv−y^3 v^3 dy = 0   ⇔ dv = y v^3  dy ; (dv/v^3 ) = y dy   ∫ (dv/v^3 ) = ∫ y dy ; −(1/(2v^2 )) = (1/2)y^2 +c   (y^2 /x^2 ) = −y^2 +C ; [ −2c = C ]  y^2 ((1/x^2 )+1) = C ; y^2  = ((Cx^2 )/(x^2 +1))  y = ± ((λx)/( (√(x^2 +1)))) ; [ λ = (√C) ]
(1)ydx(x3y+x)dy=0letx=yvdx=vdy+ydvy(vdy+ydv)(y3v3+yv)dy=0y2dvy3v3dy=0dv=yv3dy;dvv3=ydydvv3=ydy;12v2=12y2+cy2x2=y2+C;[2c=C]y2(1x2+1)=C;y2=Cx2x2+1y=±λxx2+1;[λ=C]

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