The differential equation $x \frac{d y}{d x}-y=x^2$ has the general solution:

$y=x^2+c x$, where c is a constant.

The correct answer is Option (3) → $y=x^2+c x$, where c is a constant.

$x \frac{dy}{dx}-y=x^2⇒\frac{dy}{dx}-\frac{y}{x}=x$

$I.F.:e^{\int-\frac{1}{x}dx}=e^{-\log x}=\frac{1}{x}$

so multiplying eq. by $\frac{1}{x}$

$⇒\frac{1}{x}\frac{dy}{dx}-\frac{y}{x^2}=1$

so $\frac{y}{x}=\int 1dx⇒y=x(x+c)$

$y=x^2+cx$