Solve the following differential equation: `(x^2dy)/(dx)=x^2+x y+y^2`

Explanation

Given that, `" "x^(2)(dy)/(dx)=x^(2)+xy+y^(2)`
`rArr" "(dy)/(dx)=1+(y)/(x)+(y^(2))/(x^(2))" "...(i)`
Let `" "f(x,y)=1+(y)/(x)+(y^(2))/(x^(2))`
`" "f(lamdax,lamday)=1+(lamday)/(lamdax)+(lamda^(2)y^(2))/(lamda^(2)x^(2))`
`" "f(lamdax, lamday)=lamda^(0)(1+(y)/(x)+(y^(2))/(x^(2)))`
`" "=lamda^(0)f(x,y)`
which is homogeneous expression of degree 0.
Put `" "y=vxrArr=(dy)/(dx)=v+x(dv)/(dx)`
On substituting these values in Eq. (i), we get
`" "(v+x(dv)/(dx))=1+V+V^(2)`
`rArr" "x(dv)/(dx)=1+v+v^(2)-v`
`rArr" "x(dv)/(dx)=1+v^(2)`
`rArr" "(dv)/(1+v^(2))=(dx)/(x)`
On integrating both sides, we get
`" "tan^(-1)v=log|x|+C`
`rArr" "tan^(-1)((y)/(x))=log|x|+C`