Find the curve satisfying the equation `(dy)/(dx)=(y(x+y^3))/(x(y^3-x))` and passing through the point `(4,-2)` is

Explanation

`(xy^(3)-x^(2))dy-(xy+y^(4))dx=0`
or `y^(3)(xdy-ydx) -x(xdy+ydx)=0`
or `x^(2)y^(3)(xdy-ydx)/(x^(2))-x(xdy+ydx)=0`
or `x^(2)y^(3)d(y/x) -xd(xy)=0`
Dividing by `x^(3)y^(2)`, we get
`y/xd(y/x) -(d(xy))/(x^(2)y^(2))=0`
Now, integrating, we get `1/2(y/x)^(2)+1/(xy)=c`
It passes through the point `(4,-2)`. Thus,
`=1/8-1/8=c` or `c=0`
`therefore y^(3)=-2x`