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

Explanation

Given, `x^(2)(dy)/(dx)=x^(2)+xy+y^(2)`
`:.(dy)/(dx)=1+(y)/(x)+(y^(2))/(x^(2))" "……..(i)`
`:.(dy)/(dx)=o1+(y)/(x)+((y)/(x))^(2)" "......(ii)`
Let `(y)/(x)=u`
`impliesy=u.x`
`implies(dy)/(dx)=u+x.(du)/(dx)`
`:.` Equation (i) can be written as
`u+x.(du)/(dx)=1+u+u^(2)`
`:.(du)/(1+u^(2))=(dx)/(x)`
On integrating both sides, we get
`int(du)/(1+u^(2))=int(dx)/(x)`
`:.tan^(-1)u=log|x|+c`
`:.tan^(-1)((y)/(x))=log|x|+c`