Question:
Let us consider a curve, $y=f(x)$ passing through the point $(-2,2)$ and the slope of the tangent to the curve at any point $(x, f(x))$ is given by $f(x)+x f^{\prime}(x)=x^{2}$. Then :
Correct Option: , 3
Solution:
$\mathrm{y}+\frac{\mathrm{xdy}}{\mathrm{dx}}=\mathrm{x}^{2}$ (given)
$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}+\frac{\mathrm{y}}{\mathrm{x}}=\mathrm{x}$
If $=e^{\int \frac{1}{x} d x}=x$
Solution of DE
$\Rightarrow \mathrm{y} \cdot \mathrm{x}=\int \mathrm{x} \cdot \mathrm{x} \mathrm{dx}$
$\Rightarrow \mathrm{xy}=\frac{\mathrm{x}^{3}}{3}+\frac{\mathrm{c}}{3}$
Passes through $(-2,2)$, so
$-12=-8+c \Rightarrow c=-4$.
$\therefore 3 x y=x^{3}-4$
ie. $3 x \cdot f(x)=x^{3}-4$