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

Explanation

Given equation : `x^{2} \frac{d y}{d x}=x^{2}+x y+y^{2}`

put `y=v x`

Differentiating w.r.t. `x`, we get,

$$ \Rightarrow \frac{d y}{d x}=v+x \frac{d v}{d x} $$
$$
\Rightarrow x^{2}\left[v+x \frac{d v}{d x}\right]=x^{2}\left[1+v+v^{2}\right]
...