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

Explanation

It is given that,
$$
\begin{aligned}
&x^{2} \frac{d y}{d x}=x^{2}-2 y^{2}+x y \\
&\frac{d y}{d x}=1-2 \frac{y^{2}}{x^{2}}+\frac{y}{x} \\
&\text { substitute } y=v x \rightarrow \frac{d y}{d x}=x \frac{d v}{d x}+v \\
&x \frac{d v}{d x}+v=1-2 \frac{x^{2} v^{2}}{x^{2}}+\frac{x v}{x} \\
&x \frac{d v}{d x}+v=1-2 v^{2}+v \\
...