Solve `((dy)/(dx))+(y/x)=y^3`

Explanation

Dividing the given equation by `y^(3)`, we get
`1/y^(3)(dy)/(dx)+1/y^(2)1/x=1`
Putting, `1//y^(2)=v`, we have
`(-2//y^(3))dy//dx=dv//dx`
Therefore, equation (1) becomes
`-1/2(dv)/(dx) + 1/xv=1` or `(dv)/(dx) - 2/xv=-2`
This is a linear equation with v as the dependent variable.
I.F. `e^(-int(2//x)dx)=e^(-2logx) = 1//x^(2)`
Therefore, the solution is
`v(1//x^(2)) = -2int(1//x^(2))dx+c=2//x+c`
or `2xy^(2)+cx^(2)y^(2)=1`