Find the genergal solution of `y^(2)dx+(x^(2)-xy+y^(2))dy=0`

Explanation

Given, differential equation is
`" "y^(2)dx+(x^(2)-xy+y^(2))dy=0`
`rArr" "y^(2)dx=-(x^(2)-xy+y^(2))dy`
`rArr" "y^(2)(dx)/(dy)=-(x^(2)-xy+y^(2))`
`rArr" "(dx)/(dy)=-((x^(2))/(y^(2))-(x)/(y)+1)" "...(i)`
which is a homogeneous differential equation.
Put `" "(x)/(y)=v or x=vy`
`rArr" "(dx)/(dy)=v+y(dv)/(dy)`
On substituting these values in Eq. (i), we get
`" "v+y(dv)/(dy)=-[v^(2)-v+1]`
`rArr" "y(dv)/(dy)=-v^(2)+v-1-v`
`rArr" "y(dv)/(dy)=-v^(2)-1rArr(dv)/(v^(2)+1)=-(dy)/(y)`
On integrating both sides, we get
`" "tan^(-1)(v)=-logy+C`
`rArr" "tan^(-1)((x)/(y))+logy=C" "[because v=(x)/(y)]`