x3y3(2ydx+xdy)-(5ydx+7xdy)=0

Question

solve this equation

Answer 1

x^3y^3(2ydx+xdy)-(5ydx+7xdy)=0

x^3y^3(3ydx/2+ydx/2+3xdy/2-xdy/2)-(6ydx-ydx+6xdy+xdy)=0

3x^3y^3(ydx+xdy)/2+x^3y^3(ydx-xdy)/2-6(ydx+xdy)-(ydx-xdy)=0

3x^3y^3d(xy)/2-6d(xy)+(ydx-xdy)(x^3y^3/2-1)=0

Let u=(x+y)/(x-y), then du/dx=((x-y)(1+dy/dx)-(x+y)(1-dy/dx))/(x-y)^2=

((x+xdy/dx-y-ydy/dx)-(x-xdydx+y-ydy/dx))/(x-y)^2=

(2xdy/dx-2y)/(x-y)^2=2(xdy/dx-y)/(x-y)^2, which can be written:

du=2(xdy-ydx)/(x-y)^2, from which ydx-xdy=-(x-y)^2du/2.

Let u=xy, then du/dx=xdy/dx+y or du=xdy+ydx.

Let v=x/y, then dv/dx=(y-xdy/dx)/y^2 or y^2dv=(ydx-xdy)

3u^3du/2-6du+(ydx-xdy)(u^3/2-1)=0=3u^3du/2-6u+dv/y^2(u^3/2-1)

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