9. differential equations — ગણિત ધોરણ 12 સાયન્સ — Question

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

$\Rightarrow \frac{d y}{d x}-\frac{y}{x}=2 x$

This is a linear differential equation of the form:

$\frac{d y}{d x}+p y=Q \quad$ (Where $p=-\frac{1}{x}$ and $Q=2 x$ )

The integrating factor ( $\mathrm{I.F}$ ) is given by the relation,

$e^{\int p \, d x}$

$\therefore$ $\mathrm{I.F.}$ $=e^{\int \frac{1}{x} d x}=e^{-\log x}=e^{\log \left(x^{-1}\right)}=x^{-1}=\frac{1}{x}$

Hence, the correct answer is $\mathrm{A}.$