The differential equation $x \frac{d y}{d x}-y=x^2$ has the general solution:
Answer & explanation
Correct answer: option 3
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 $\frac{y}{x}=\int 1dx⇒y=x(x+c)$
$y=x^2+cx$